FG, GH, FH, F, G, H Warm Up 1. Name all sides and angles of ∆FGH. 2. What is true about K and L? Why? 3. What does it mean for two segments to be congruent? FG, GH, FH, F, G, H ;Third s Thm. They have the same length.
Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence.
The Idea of a Congruence Two geometric figures with exactly the same size and shape. A C B D E F Each triangle is made up of six parts: 3 sides and 3 angles
Corresponding Parts If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B A C AB DE BC EF AC DF A D B E C F ABC DEF E D F
For example, P and Q are consecutive vertices. P and S are not. Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. P and S are not. Helpful Hint P Q S
To name a polygon, write the vertices in consecutive order. In a congruence statement, the order of the vertices indicates the corresponding parts. so the order MATTERS!
When you write a statement such as ABC DEF, you are also stating which parts are congruent. Helpful Hint
Identify all pairs of corresponding congruent parts. 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
CPCTC Corresponding Parts of Congruent Triangles are Congruent.
Congruence Transformations Congruency amongst triangles does not change when you… slide (translation), Turn (rotation), or flip (reflection) … the triangles.
EXAMPLE 1 Identify congruent parts Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.
Complete each congruence statement.
Complete each congruence statement.
Write a congruence statement.
Write a congruence statement.
Given ΔBCD = ΔRST, m C = 57, m R = 64, m D = 5x + 4. Find x and m T. Given ΔQRS = ΔLMN, RS = 8, QR=14, QS=10, MN = 3x – 25. Find x.
Given ΔBCD = ΔRST, m C = 57, m R = 64, m D = 5x + 4. Find x and m T. Given ΔQRS = ΔLMN, RS = 8, QR=14, QS=10, MN = 3x – 25. Find x.
Name the corresponding angles and sides if ΔBCD = ΔEFG. Given ΔCAT = ΔDOG, CA=14, AT=18, TC=21 and DG=2x+7, find the value of x.
Name the corresponding angles and sides if ΔBCD = ΔEFG. Given ΔCAT = ΔDOG, CA=14, AT=18, TC=21 and DG=2x+7, find the value of x.
Given: ∆ABC ∆DBC. BCA and BCD are rt. s. Def. of lines. Example 2A: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find the value of x. Find mDBC. BCA and BCD are rt. s. Def. of lines. BCA BCD Rt. Thm. mBCA = mBCD Def. of s Substitute values for mBCA and mBCD. (2x – 16)° = 90° 2x = 106 Add 16 to both sides. x = 53 Divide both sides by 2.
Given: ∆ABC ∆DBC. ∆ Sum Thm. mABC + mBCA + mA = 180° Example 2B: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find mDBC. mABC + mBCA + mA = 180° ∆ Sum Thm. mABC + 90 + 49.3 = 180 Substitute values for mBCA and mA. mABC + 139.3 = 180 Simplify. mABC = 40.7 Subtract 139.3 from both sides. DBC ABC Corr. s of ∆s are . mDBC = mABC Def. of s. mDBC 40.7° Trans. Prop. of =
Given: ∆ABC ∆DEF AB DE Corr. sides of ∆s are . AB = DE Check It Out! Example 3a Given: ∆ABC ∆DEF Find the value of x. Find mF. 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.
Given: ∆ABC ∆DEF ∆ Sum Thm. mEFD + mDEF + mFDE = 180° ABC DEF Check It Out! Example 3b Given: ∆ABC ∆DEF Find mF. ∆ Sum Thm. mEFD + mDEF + mFDE = 180° ABC DEF Corr. s of ∆ are . mABC = mDEF Def. of s. mDEF = 53° Transitive Prop. of =. Substitute values for mDEF and mFDE. mEFD + 53 + 90 = 180 mF + 143 = 180 Simplify. mF = 37° Subtract 143 from both sides.
1. ∆ABC ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB. Lesson Quiz 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 ____