**CHAPTER**

**Triangle Congruence**

**3**

**Lesson 3.1**

__Conditions for Triangle Congruence__(Textbook pages 204-226, 3 sessions)

**I.**

**Mathematical Concepts and Skills**

A. Two triangles are congruent if their corresponding parts are congruent.

B. Congruence for triangles is

*reflexive, symmetric,*and*transitive.*C. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

D. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

E. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

F. If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, respectively, then the triangles are congruent.

G. The SSS Postulate, SAS Postulate, ASA Postulate, and the SAA Theorem are used to prove that two triangles are congruent.

**II.**

**Objectives**

A. Illustrate and define congruent triangles.

B. Prove that congruence for triangles is reflexive, symmetric, and transitive.

C. Use inductive skills to establish the conditions or correspondence sufficient to guarantee congruence between triangles.

D. Apply deductive skills to show congruence between triangles.

**III.**

**Values Integration**

The need to be honest and the importance of curiosity.

**IV.**

**Materials**

bond paper ruler

Pair of scissors protractor

**V.**

**Instructional Strategies**

A. Whole Class Discussion

B. Practical Work

**VI.**

**Procedure**

**Session 1**

**A.**

**Preliminary Activities**

1.Check the assignment.

2.Recall the definition of congruent triangles.

3.Recall Theorem 2-3: Congruence for angles is an equivalence

**B.**

**Lesson Proper**

1. Ask the students to draw any triangle. Tell them to cut out their paper triangle and then paste it on two sheets of bond paper.

2. Ask the students to compare the two triangles. If they find difficulties comparing the two triangles, tell them to do the following. Tell them also that being curious is very important in mathematics.

a. measure their angles using protractor

b. measure their sides using a ruler

3. Introduce the terms

*corresponding angles, corresponding sides, and congruent triangles*including the symbol “↔” for*correspondence.*4. Ask what makes two triangles congruent.

5. Prove Theorem 3-1.

6. Tell the students to work with a partner and work on Explore on page 208 of the textbook.

7. Practice Exercises

a. Answer Mental Mathematics of Exercise 3.1 (numbers 1-5) on page 218 of the textbook.

b. Solve Written Mathematics of Exercise 3.1 (numbers 1 and 2) on pages 220-221 of the textbook.

**C. Assignment**

**Solve Written Mathematics of Exercise 3.1 (numbers 3-7) on pages 221-222 of the textbook.**

**Session 2**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the three Congruence Postulates for Triangles.

**B.**

**Follow-up Lesson**

1. Illustrative Examples

Use the example on page 212 of the textbook.

2. Explain the proof of SAA Theorem.

3. Practice Exercises

Solve Written Mathematics of Exercise 3.1 (numbers 8-11) on page 223 of the textbook.

**C.**

**Assignment**

**Solve Written Mathematics of Exercise 3.1 (numbers 12 and 13) on page 224 of the textbook.**

**Session 3**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the SAA Theorem.

**B.**

**Follow-up Lesson**

1. Illustrative Examples

Use the example on page 213 of the textbook.

2. Practice Exercises

Solve Written Mathematics of Exercise 3.1 (numbers 14-16) on page 224 of the textbook.

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 3.1 (numbers 17 and 18) on page 225 of the textbook. Cover your work to be honest.

**D.**

**Assignment**

Solve Written Mathematics of Exercise 3.1 (numbers 19 and 20) on page 225 of the textbook.

**Lesson 3.2**

__Applying the Conditions for Triangle Congruence__(Textbook pages 227-238, 2 sessions)

**I.**

**Mathematical Concepts and Skills**

A. The corresponding parts of two triangles can be approved congruent by using the definition of congruent triangles, the congruence postulates for triangles, and the SAA Theorem.

B. Every angle has exactly one bisector.

**II.**

**Objectives**

A. To prove the corresponding parts of two congruent triangles are congruent.

B. To apply the congruence postulates in proving triangles are congruent.

**III.**

**Values Integration**

Tolerance for the feelings, actions, and attitudes of others

**IV.**

**Materials**

manila paper marker

**V.**

**Instructional Strategies**

A. Discussion

B. Discovery

**VI.**

**Procedure**

**Session 1**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the following:

a. SSS Postulate

b. SAS Postulate

c. ASA Postulate

d. SAA Postulate

**B. Lesson Proper**

1. Present the following triangles on the board:

2. Ask the students to complete the following:

a. AB is congruent to ___

b. BC is congruent to ___

c. AC is congruent to ___

d. <A is congruent to ___

e. <B is congruent to ___

f. <C is congruent to ___

3. The students should see that when two triangles are congruent their corresponding sides and corresponding angles are also congruent.

4. Give illustrative examples on how to prove the corresponding parts of congruent triangles are congruent.

5. Prove the Angle Bisector Theorem.

6. Practice Exercises.

a. Answer the Mental Mathematics of Exercise 3.2 (numbers 1-4) on page 232 of the textbook.

b. Solve Written Mathematics of Exercise 3.2 (numbers 1-3) on page 233 of the textbook.

**C. Assignment**

**Solve Written Mathematics of Exercise 3.2 (numbers 4-8) on page 234 of the textbook.**

**Session 2**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the definition of angle bisector and the definition of perpendicular.

**B.**

**Follow-up Lesson**

1. Illustrative Example

Use the example on page 231 of the textbook.

2. Practice Exercises

Solve Written Mathematics of Exercise 3.2 (numbers 9-12) on page 236 of the textbook.

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 3.2 (numbers 13-16) on page 236 of the textbook.

Tell the students to be honest in doing the exercises.

**D.**

**Assignment**

Solve Written Mathematics of Exercise 3.2 (numbers 17-20) on 237 for the textbook.

**If there is enough time, solve the Challenge Problems of Exercise 3.2 on page 238 of the textbook.**

**Lesson 3.3**

__Congruence and Inequality Properties in an Isosceles Triangle__(Textbook pages 239-252, 3 sessions)

**I.**

**Mathematical Concepts and Skills**

A. If two sides of a triangle are congruent, then the angles opposite these sides.

B. Every equilateral triangle is equiangular.

C. Each angle of an equilateral triangle has a measure of 60°.

D. If two angles of a triangle are congruent, then the sides opposite these angles are congruent.

E. Every equiangular triangle is equilateral.

F. If one side of the triangle is longer than the second side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side.

G. If one angle of the triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

H. If two sides of a triangle are congruent to two sides of a second triangle and the included angle of the first triangle has a greater measure than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

I. If two sides of one triangle are congruent to two sides of a second triangle and the included angle of the first triangle has a greater measure than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

**II. Objectives**

A. Prove and apply the Isosceles Triangle Theorem and its converse.

B. Prove that every equilateral triangle is equiangular.

C. Prove that every equiangular triangle is equilateral.

D. Prove that every angle of an equilateral has a measure of 60°.

E. Prove that if one side of a triangle is longer than a second side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side.

F. Prove that if one angle of a triangle is longer than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

G. Prove and apply the Hinge Theorem and its converse.

**III. Values Integration**

**The importance of joint effort in accomplishing a given task**

**IV. Materials**

ruler protractor straws

**V. Instructional Strategies**

A. Discussion

B. Group Work

C. Practical Work in Small Groups

**VI. Procedure**

**Session 1**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the definition of isosceles triangle.

**B.**

**Lesson Proper**

1. Remind the students about the importance of joint effort in accomplishing a given task and then tell them to do the following activity in pairs.

a. Draw an isosceles triangle

*ABC*with base*BC*.b. Using a protractor and a ruler, bisect the vertex angle

*BAC*.c. Let the bisector intersect

*BC*at point*D.*d. With a protractor measure <

*BAD*and <*CAD*.2. Ask the following questions:

a. Why are <

b. Is side

c. Is side

*BAD*and <*CAD*congruent?b. Is side

*AB*congruent to side*AC*? Why?c. Is side

*AD*congruent to itself? Why? d. Can you now conclude that ∆

*BAD*is congruent to ∆*CAD*? Why? e. Is <

*B*congruent to <*C*? Why?3. Explain the proof of the Isosceles Triangle Theorem and Theorem 3-5.

4. Practice Exercises

a. Answer Mental Mathematics of Exercise 3.3 (numbers 1-5) on page 246 of the textbook.

b. Solve Written Mathematics of Exercise 3.3 (numbers 1 and 2) on pages 247-248 of the textbook.

**C. Assignment**

Solve Written Mathematics of Exercise 3.3 (numbers 3 and 4) on page 249 of the textbook.

**Session 2**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the Angle Addition Postulate, Theorem 2-10, and Theorem 2-11.

**B.**

**Follow-up Lesson**

1. Divide the students in small groups. Tell them to do the following activities.

Activity 1

a. Draw a scalene triangle.

b. Measure each side of the triangle in centimetres.

c. Label the longest side AB

*and the shortest side BC*d. Measure the angle opposite to AB and the angle opposite AC. Compare their measurements

e. Measure the angle opposite to BC. Compare its measure with the measure of the angle opposite AB.

f. Compare the measurement to the angle opposite AC with the measure of the angle opposite BC.

g. What can you conclude?

**Activity 2**

a. Draw a scalene triangle.

b. Label the angle which seems to have the largest measure A, and the angle which seems to have the smallest measure B.

c. Measure each angle and the side opposite each angle.

d. What can you conclude?

2. Complete the proof of the Angle Side Inequality Theorem.

3. Explain the proof of Theorem 3-6: The Side Angle Inequality Theorem.

4. Illustrative Example

Use the examples on page 244 of the textbook.

5. Practice Exercises

Solve Written Mathematics of Exercise 3.3 (number 5) on page 249 of the textbook.

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 3.3 (numbers 11-16) on the page 250 of the textbook.

D.

**Assignment**Solve Written Mathematics of Exercise 3.3 (numbers 17-20) on page 251.

**If there is enough time, solve the Challenge Problems of Exercise 3.3 on page 252 of the textbook.**

**Lesson 3.4**

**Overlapping Triangles**

(textbook pages 253-261, 1 session)

**I.**

**Mathematical Concepts and Skills**

A. A geometric figure may be made up of two or more overlapping triangles.

B. To prove overlapping triangles, it is better to mentally separate or redraw into separate figures.

**II.**

**Objective**

To prove that overlapping triangles are congruent

**III.**

**Values Integration**

The value of hard-work

**IV.**

**Materials**

Wires

Rulers

**V.**

**Instructional Strategies**

A. Practical Work

B. Discussion

**VI.**

**Procedure**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the Three Congruence Postulate for Triangles.

**B.**

**Lesson Proper**

1. Using wires, make two congruent triangles.

2.

Let two triangles overlap.

Let two triangles overlap.

After doing the activity, ask the students whether they find it easy to make two congruent triangles using wires. Tell them t hat life is not always a bed of roses. One must work hard to succeed in life.

3.

Write a problem using a figure similar to the overlapping triangles made of wires.

Write a problem using a figure similar to the overlapping triangles made of wires.

**A D**Given: AB ≈ DC

AC ≈ DB

Prove: ΔABC ≈ ΔDCB

**B C**

4. Tell the students to redraw the overlapping triangles into separate figure. Then ask them to write a proof.

5. Practice Exercise

a. Answer the Mental Mathematics of Exercise 3.4 (numbers 1-4)

on page 255 of the textbook.

b. Solve Written Mathematics of Exercise 3.4 (numbers 1-4) on

on pages 256-257

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 3.4 (numbers 5-8) on page 258 of the textbook.

**D.**

**Assignment**

Solve Written Mathematics of Exercise 3.4 (numbers 9-13) on page 259 of the textbook.

**Lesson 3.5**

**Right Triangle Congruence and Problems Involving Altitudes and Medians**

(Textbook pages 262-275, 3 sessions)

**I.**

**Mathematical Concepts and Skills**

A. If two legs of a right triangle are congruent to the corresponding leg of another right triangle, then two triangles are congruent.

B. If one leg and one acute angle of a right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the two right triangles are congruent.

C. If the hypotenuse and one leg of a right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the two right triangles are congruent.

D. If the hypotenuse and one acute angle of a right triangle mare congruent to the corresponding hypotenuse and acute angle of another right triangle, then the two right triangles are congruent.

**II.**

**Objectives**

A. Identify the parts of a right triangle

B. Prove and apply the following theorems:

a. Leg-Leg Congruence Theorem

b. Leg-Acute Angle Congruence Theorem

c. Hypotenuse-Leg Congruence Theorem

d. Hypotenuse-Acute Angle Congruence Theorem

C. Solve problems involving altitudes and medians

**III.**

**Values Integration**

The need to act with prudence

**IV.**

**Materials**

Manila paper

Marker

**V.**

**Instructional Strategy**

Discussion

**VI.**

**Procedure**

**Session 1**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the SAS Postulate.

**B.**

**Lesson Proper**

1. Draw a right triangle on a manila paper. Post the manila paper on the board. Ask one student to identify its parts.

2. Ask the students to enumerate all the postulates, definitions, and theorems that can be used to prove that two triangles are congruent.

3. Guide the students in proving the Leg-Leg Congruence Theorem and the Hypotenuse-Leg Congruence Theorem.

4. Practice Exercises

Prove the Leg-Acute Angle Congruence Theorem and the Hypotenuse-Acute Angle Congruence Theorem. Tell the students that they should be careful in writing the proof to avoid erroneous steps and reasons. Even in life, it is always better to act with prudence.

**C.**

**Assignment**

Solve Written Mathematics of Exercise 3.5 (numbers 1-5) on page 272 of the textbook.

**Session 2**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the Isosceles Triangle Theorem, the SAS, ASA, SSS Postulates, and the SAA Theorem.

**B.**

**Follow-up Lesson**

1. Illustrative Examples

Use the examples on page 263 of the textbook.

2. Practice Exercises

a. Answer the Mental Mathematics of Exercise 3.5 (numbers 1-6) on page 271 of the textbook.

b. Solve Written Mathematics of Exercise 3.5 (numbers 1-5) on page 272 of the textbook.

**C.**

**Assignment**

Solve Written Mathematics of Exercise 3.5 (numbers 6-4) on page 273 of the textbook.

**Session 3**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall the definition of altitude and definition of median.

**B.**

**Follow-up Lesson**

1. Illustrative Examples

Use the example on page 268 of the textbook.

2. Practice Exercises

Solve Written Mathematics of Exercise 3.5 (numbers 10-13) on page 274 of the textbook.

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 3.5 (numbers 14-15) on page 274 of the textbook.

**D.**

**Assignment**

Solve Written Mathematics of Exercise 3.5 (numbers 16-18) on page 275 of the textbook.

**If there is enough time, solve the Challenge Problems of Exercise 3.5 on page 275 of the textbook.**

**Lesson 3.6**

**Theorems Related to Bisectors and Perpendicular Lines**

(Textbook pages 276-288, 2 sessions)

**I.**

**Mathematical Concepts and Skills**

A. The distance from a point to a line is the length of the perpendicular segment from the point of the line.

B. A point on the bisector of an angle is equidistant from the sides of the angle.

C. A point equidistant from the sides of an angle lies on the bisector of an angle.

D. A point on the perpendicular bisector of a segment is equidistant from the endpoint of a segment.

E. A point equidistant from the endpoint of a segment lies on the perpendicular bisector of the segment.

F. If a line contains two points each of which is equidistant from the endpoints of a segment, then the line is the perpendicular bisector of the segment.

G. On a plane through a given point on a line, there is exactly one line perpendicular to the line.

H. There is one and only one line perpendicular to a given line from an external point.

**II.**

**Objectives**

A. Define the distance from a point to a line

B. Prove and apply the theorems about bisectors and perpendicular lines

**III.**

**Value Integration**

Practicality

**IV.**

**Materials**

Coin

Ruler or tape measure

**V.**

**Instructional Strategies**

A. Whole Class Discussion

B. Practical Work

**VI.**

**Procedure**

**Session 1**

**A.**

**Preliminary Activities**

1. Check the Assignment.

2. Recall the definition of perpendicular

**B.**

**Lesson Proper**

1. Put a coin on the floor. Have one student measure the distance from one coin to one of the walls of the room.

2. Draw a line and external point on the board. Ask one student to determine the distant from the point to the line.

3.

Ask another student to draw a perpendicular segment from the external point to the line. Then tell the student to measure the length of the segment.

Ask another student to draw a perpendicular segment from the external point to the line. Then tell the student to measure the length of the segment.

Ask whether the length of the segment is equal to the distance from the external point to the line.

4. Ask the students to define the distance from an external point to the line. Ask them to relate the activity to something they have done before. Did they use their past experiences in doing the task? Ask them the importance of practicality in life.

5. Discuss the proof of Theorem 3-14 and Theorem 3-16

6. Practice Exercise

Prove Theorem 3-15

**C.**

**Assignment**

1. Study the proof of Theorem 3-17, Corollary 3-17.1, and Theorem 3-18

2. Prove Theorem 3-19

**Session 2**

**A.**

**Preliminary Activities**

1. Check the assignment

2. Recall the definition of a linear pair and the Linear Pair Postulate

**B.**

**Follow-up Lesson**

1. Prove Theorem 3-17, Corollary 3-17.1, and Theorem 3-18

2. Illustrative Example

Explain the proof of this problem.

Given: AD ┴ bisector of BC

Prove: ΔABC is isosceles

**A**

**B D C**

3. Practice Exercises

a. Answer Mental Mathematics of Exercise 3.6 (numbers 1-5) on page 284 of the textbook.

b. Solve Written Mathematics of Exercise 3.6 (number 1-6) on page 285 of the textbook.

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 3.6 (numbers 7-10) on page 286-287 of the textbook.

**If there is enough time, solve the Challenge Problems of Exercise 3.6 on pages 288 of the textbook.**

**Lesson 4.1**

Types of Quadrilaterals and their Properties

(Textbook pages 296-307, 3 sessions)

I.

**Mathematical Concepts and Skills**A. A diagonal of a parallelogram forms two congruent triangles.

B. In a parallelogram, any two opposite angles are congruent

C. In a parallelogram, any two consecutive angles are supplementary

D. In a parallelogram, any two opposite sides are congruent

E. The diagonals of a parallelogram bisect each other

F. If two lines are parallel, then all points of each line are equidistant from the other line.

**II. Objectives**

A. Recall previous knowledge on the different types of quadrilaterals

B. Apply inductive and deductive skills to derive the properties of a parallelogram

C. Apply the theorems about the properties of parallelograms

**III. Values Integration**

Carefulness in accomplishing a task

**IV. Materials**

geoboard

rubber bands

**V. Instructional Strategies**

A. Practical Work

B. Discussion

**VI. Procedure**

**Session 1**

**A.**

**Preliminary Acqtivities**

1. Check the Assignment

2. Recall the types of quadrilaterals

**B.**

**Lesson Proper**

1. Tell the students to do the following activity. Remind the students to be always careful in doing their task

a. Using a geoboard and rubber bands, make quadrilaterals with the following descriptions:

· A quadrilateral in which both pairs of opposite sides are parallel

· A parallelogram with four right angles

· A parallelogram with four congruent sides

· A rectangle with four congruent sides

· A quadrilateral with exactly one pair of parallel sides

b. Each time you are finished making one of the quadrilaterals, ask your seatmate these questions: “What type of quadrilateral is on my geoboard? Why?”

2. Ask the students to work on Explore on pages 297-298 of the textbook.

3. Discuss the proofs of Theorems 4-1, 4-2, 4-3. 4-4. And 4-5

4. Practice Exercises

a. Answer Mental Mathematics of Exercise 4.1 (numbers 1-6) on page 303 of the textbook

b. Prove Corollary 4-1.1

**C.**

**Assignment**

Solve Written Mathematics of Exercise 4.1 (numbers 1-4) on page 304 of the textbook.

**Session 2**

**A.**

**Preliminary Activities**

1. Check the assignment

2. Recall the Systems of Linear Equations

**B.**

**Follow-up Lesson**

1. Illustrative Examples

Explain the proof of this problem

Given: Quadrilateral ABCD is a parallelogram

1 ≈ 2

Prove: 3 ≈ 4

2. Practice Exercises

Solve Written Mathematics of Exercise 4.1 (numbers 5-7) on page 304 of the textbook

**C.**

**Assignment**

Solve Written Mathematics of Exercise 4.1 (numbers 8-10) on page 305 of the textbook.

**Session 3**

**A.**

**Preliminary Activities**

1. Check the assignment.

2. Recall Theorems 4-1 to 4-5

**B.**

**Follow-up Lesson**

1. Illustrative Examples

Given: Quadrilateral CDEF is a parallelogram

FH = DH

Prove: Δ GDE ≈ ΔHOC

2. Practice Exercises

Solve Written Mathematics of Exercise 4.1 (numbers 11-13) page 305 of the textbook.

**C.**

**Checking for Understanding**

Solve Written Mathematics of Exercise 4.1 (numbers 16 and 17) on page 306 of the textbook.

**If there is enough time, solve Challenge Problems of Exercise 4.1 on page 307 of the textbook.**

**Lesson 4.2**

**Properties of the Diagonals of Special Quadrilaterals**

**(**Textbook pages 308-314, 2 sessions)

**I. Mathematical Concepts and Skills**

A. The diagonals of a rectangle are congruent

B. The diagonals of a square bisect the vertex angle

C. The diagonals of a rhombus are perpendicular

D. Each diagonal of a rhombus, bisects the opposite angles of the rhombus

E. If a parallelogram has at least one right angle, then it is a rectangle

F. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus

**II. Objectives**

A. Apply inductive and deductive skills to derive the additional properties of rectangles, rhombi, and squares.

B. Apply the Theorems about rhombi, rectangles and squares

**III. Values Integration**

The importance of cooperation in accomplishing a task.

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