1. 4-4 Congruent Triangles Warm Up Lesson Presentation Lesson Quiz
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry

2. 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.

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

4. 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.

6. 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

7. 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!

8. When you write a statement such as ABC  DEF, you are also stating which parts are congruent.
Helpful Hint

9. 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

10. 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.

11. 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 =

12. 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.

13. 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.

14. OUR FIRST PROOF
is here!!!!!
yay

15. 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.)

16. 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”

17. 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.

18. 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

19. 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

20. 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

21. 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

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

23. 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.

24. 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  ∆

25. 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.

26. 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

27. Lesson Quiz
1. ∆ABC  ∆JKL and AB = 2x JK = 4x – 50. Find x and AB.
Given that polygon MNOP  polygon QRST, identify the congruent corresponding part.
2. NO  ____ T  ____
4. Given: C is the midpoint of BD and AE.
A  E, AB  ED
Prove: ∆ABC  ∆EDC
31, 74 RS P

28. 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