1. Objectives Use properties of congruent triangles.
Prove triangles congruent by using the definition of congruence.

2. Geometric figures are congruent if they are the same size and shape
Geometric figures are congruent if they are the same size and shape. Corresponding angles and 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. For example, P and Q are consecutive vertices.
Two vertices that are the endpoints of a side are called consecutive vertices.
For example, P and Q are consecutive vertices.
Helpful Hint

5. To name a polygon, write the vertices in consecutive order
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.

6. 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

7. Check It Out! Example 1
If polygon LMNP  polygon EFGH, identify all pairs of corresponding congruent parts.
Angles: L  E, M  F, N  G, P  H
Sides: LM  EF, MN  FG, NP  GH, LP  EH

8. Example 2A: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find the value of x.
BCA and BCD are rt. s.
Def. of  lines.
BCA  BCD
Rt.   Thm.
mBCA = mBCD
Def. of  s
Substitute values for mBCA and mBCD.
(2x – 16)° = 90°
2x = 106
Add 16 to both sides.
x = 53
Divide both sides by 2.

9. Example 2B: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find mDBC.
∆ Sum Thm.
mABC + mBCA + mA = 180°
Substitute values for mBCA and mA.
mABC = 180
mABC = 180
Simplify.
Subtract from both sides.
mABC = 40.7
DBC  ABC
Corr. s of  ∆s are  .
mDBC = mABC
Def. of  s.
mDBC  40.7°
Trans. Prop. of =

10. Check It Out! Example 2a Given: ∆ABC  ∆DEF Find the value of x.
AB  DE
Corr. sides of  ∆s are .
AB = DE
Def. of  parts.
Substitute values for AB and DE.
2x – 2 = 6
2x = 8
Add 2 to both sides.
x = 4
Divide both sides by 2.

11. Check It Out! Example 2b Given: ∆ABC  ∆DEF Find mF. ∆ Sum Thm.
mEFD + mDEF + mFDE = 180°
ABC  DEF
Corr. s of  ∆ are .
mABC = mDEF
Def. of  s.
mDEF = 53°
Transitive Prop. of =.
Substitute values for mDEF and mFDE.
mEFD = 180
mF = 180
Simplify.
mF = 37°
Subtract 143 from both sides.