4-4 Congruent Triangles Warm Up Lesson Presentation Lesson Quiz Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry

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.

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.

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

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

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.

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 + 90 + 49.3 = 180 mABC + 139.3 = 180 Simplify. Subtract 139.3 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 =

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.

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 + 53 + 90 = 180 mF + 143 = 180 Simplify. mF = 37° Subtract 143 from both sides.

OUR FIRST PROOF is here!!!!! yay

There are two ways to format proofs Way 1: Two column proof (this is the way we will do our proofs… College professors everywhere will laugh at us, but we don’t care!) Way 2: Paragraph proof (You can choses to do you proofs this way and impress the laughing college professors, but you don’t have to succumb to their mockery.)

A two-column proof has…surprise… TWO columns… Statements Reasons “Word” stuff “Math” stuff You will always be given 1 or more “Givens” and you will always be given a “Prove”

Hmmmm….. In order to “prove” that two triangles are congruent, what do we need…?? Hmmmm….. We must know that all corresponding angles are congruent, and all corresponding sides are congruent. SO that is 6 things we need to put in our proof.

Step 1: MARK IT UP!!! Step 2: Decide what you are using Step 3: ATTACK! Check off the use Step 4: Get to the end goal, the PROVE

Example 3: Proving Triangles Congruent Given: YWX and YWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY  YZ. Prove: ∆XYW  ∆ZYW Start with step 1

So lets talk about what we have marked up, and what we do not have marked up. 2 sides congruent 2 angles congruent MISSING 1 side pair 1 angle pair

What is up with the last side? YW is the same in both triangles, therefore it is congruent to itself! This is the reflexive property. We are going to use this A LOT

What is up with the last angle pair? By the third angle theorem, the last two angles must be congruent!

We officially have a reason for each side being congruent and each angle being congruent. Let’s pop it into our fancy two-column proof.

A A A S S S Statements Reasons 1. YWX and YWZ are rt. s. 1. Given 2. YWX  YWZ 2. Rt.   Thm. A 3. YW bisects XYZ 3. Given 4. XYW  ZYW 4. Def. of bisector A 5. X  Z 5. Third s Thm. A 6. W is midpoint. of XZ 6. Given 7. XW  ZW 7. Def. of midpoint S 8. YW  YW 8. Reflex. Prop. of  S 9. XY  YZ 9. Given S 10. ∆XYW  ∆ZYW 10. Def. of  ∆

MARK IT UP Check It Out! Example 3 Given: AD bisects BE. BE bisects AD. AB  DE, A  D MARK IT UP Prove: ∆ABC  ∆DEC We are missing 2 angles this time. Because they are vertical angles. By the third angle theorem.

1. A  D 1. Given A 2. BCA  DCE 2. Vertical s are . A Statements Reasons 1. A  D 1. Given A 2. BCA  DCE 2. Vertical s are . A 3. ABC  DEC 3. Third s Thm. A 4. Given 4. AB  DE S 5. Given BE bisects AD 5. AD bisects BE, 6. BC  EC, AC  DC 6. Def. of bisector S S 7. ∆ABC  ∆DEC 7. Def. of  ∆s

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  ____ 4. Given: C is the midpoint of BD and AE. A  E, AB  ED Prove: ∆ABC  ∆EDC 31, 74 RS P

Lesson Quiz 4. 7. Def. of  ∆s 7. ABC  EDC 6. Third s Thm. 6. B  D 5. Vert. s Thm. 5. ACB  ECD 4. Given 4. AB  ED 3. Def. of mdpt. 3. AC  EC; BC  DC 2. Given 2. C is mdpt. of BD and AE 1. Given 1. A  E Reasons Statements