4-1 Congruent Figures

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Lesson Presentation

Exit Ticket

4-1 Congruent Figures

Warm Up #15

Classify each triangle by its angles and sides.

1. MNQ

2. NQP

3. MNP

4. Find the side lengths of the triangle.

acute; equilateral

obtuse; scalene

acute; scalene

29; 29; 23

4-1 Congruent Figures

Solve It !!!

y◦

mA = 180 – (30 + 75)A

B

C F

E

D

30o 30o

A. What is mA?

= 180 – (105)

mA = 75o

∆ Sum Theorem

B. What is mD?

The wings of an SR-71 Blackbird aircraft suggest congruent triangles.

4-1 Congruent Figures

Connect Mathematical Ideas (1)(F)

How does this problem relate to a problem you have seen before ?

Communicate

4-1 Congruent Figures

Connect to Math

SWBAT

1. Use properties of congruent triangles.

2. Prove triangles congruent by using the definition of congruence.

By the end of today’s lesson,

4-1 Congruent Figures

corresponding angles

corresponding sides

congruent polygons

Vocabulary

4-1 Congruent Figures

Geometric figures are congruent if they are the same size and shape. Corresponding anglesand corresponding sides are in the same position in polygons with an equal number of sides.

Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

4-1 Congruent Figures

4-1 Congruent Figures

Two vertices that are the endpoints of a side are called consecutive vertices.

For example, P and Q are consecutive vertices.

Helpful Hint

4-1 Congruent Figures

To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS.

In a congruence statement, the order of the vertices indicates the corresponding parts.

4-1 Congruent Figures

When you write a statement such as ABC DEF, you are also stating which parts are congruent.

Helpful Hint

4-1 Congruent Figures

Using Congruent Sides and Angles

y◦

A

B

C F

E

D

30o 30o

A. Identify all pairs of corresponding congruent parts.

C. What is mD?

Angles: B E, C F, A D

Sides: 𝐴𝐵 ≅ 𝐷𝐸; 𝐵𝐶 ≅ 𝐸𝐹; 𝐴𝐶 ≅ 𝐷𝐹

B. What is mA? 75o

75o Corresponding angles of

congruent triangles

The wings of an SR-71 Blackbird aircraft suggest congruent triangles.

4-1 Congruent Figures

Example 1: Naming Congruent Corresponding Parts

Given: ∆PQR ∆STW

Identify all pairs of corresponding congruent parts.

Angles: P S, Q T, R W

Sides: 𝑃𝑄 ≅ 𝑆𝑇; 𝑄𝑅 ≅ 𝑇𝑊; 𝑃𝑅 ≅ 𝑆𝑊

4-1 Congruent Figures

Example 2A: Using Corresponding Parts of Congruent Triangles

Given: ∆ABC ∆DBC.

Find the value of x.

BCA and BCD are rt. s.

BCA BCD

mBCA = mBCD

(2x – 16)° = 90°

2x = 106

x = 53

Def. of lines.

Rt. Thm.

Def. of s

Substitute values for mBCA and mBCD.

Add 16 to both sides.

Divide both sides by 2.

4-1 Congruent Figures

Example 2B: Using Corresponding Parts of Congruent Triangles

Given: ∆ABC ∆DBC.

Find mDBC.

mABC + mBCA + mA = 180°

mABC + 90 + 49.3 = 180

mABC + 139.3 = 180

mABC = 40.7

mDBC = mABC

∆ Sum Thm.

Simplify.

Subtract 139.3 from both sides.

Def. of s.

mDBC 40.7° Trans. Prop. of =

Substitute values for mBCA and mA.

Know: DBC ABC Corr. s of ∆s are .

4-1 Congruent Figures

Given: ∆ABC ∆DEF

Example 3:

Find the value of x.

2x – 2 = 6

2x = 8

x = 4

Corr. sides of ∆s are .

Add 2 to both sides.

Divide both sides by 2.

𝐴𝐵 ≅ 𝐷𝐸

Substitute values for AB and DE.

AB = DE Def. of parts.

4-1 Congruent Figures

4-1 Congruent Figures

Example 4

Given: 𝐴𝐷 bisects 𝐵𝐸.

Prove: ∆ABC ∆DEC

𝐵𝐸 bisects 𝐴𝐷.

𝐴𝐵 ≅ 𝐷𝐸,A D.

5. Def. of bisector

6. Def. of ∆s6. ∆ABC ∆DEC

3. ABC DEC

4. Given

2. BCA DCE

3. Third s Thm.

2. Vertical s are .

1. Given1. 𝐴𝐵 ≅ 𝐷𝐸; A D

Statements Reasons

4. 𝐴𝐷 bisects 𝐵𝐸 ; 𝐵𝐸 bisects 𝐴𝐷

5. 𝐵𝐶 ≅ 𝐸𝐶; 𝐴𝐶 ≅ 𝐷𝐶

4-1 Congruent Figures

Example 5: Engineering Application

Given: PR and QT bisect each other.

PQS RTS, QP RT

Prove: ∆QPS ∆TRS

The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent.

Statements Reasons

1. 𝑃𝑅 and 𝑄𝑇 bisect each other.

PQS RTS, 𝑄𝑃 ≅ 𝑅𝑇

1. Given

2. Def. of bisector2. QS TS, PS RS

5. Def. of ∆s5. ∆QPS ∆TRS

4. Third s Thm.4. QSP TRS

3. Vert. s Thm.3. QSP TSR

4-1 Congruent Figures

Exit Ticket

1. ∆ABC ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB.

Given that polygon MNOP polygon QRST, identify the congruent corresponding part.

2. NO ____ 3. T ____

4. Given: C is the midpoint of BD and AE.

A E, AB ED

Prove: ∆ABC ∆EDC