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Name_________________________________
Geometry Period _____
5-3 Notes
Date______
5-3: Triangle Congruence and Proof Pieces
Learning Goals: What does it mean for 2 triangles to be congruent? What can we infer based on knowing that 2
triangles are congruent? What important aspects are included in two-column proofs?
Warm-Up
1. From our homework last night…
What are some things you can conclude from the diagrams given?
*Can we conclude that these two triangles are congruent? Why or why not?
*So, if given ΔABC ≅ ΔDEF, we can identify the pairs of congruent corresponding parts.
Corresponding Angles
Corresponding Sides
<A 
̅̅̅̅ 
𝐴𝐵
<B 
̅̅̅̅ 
𝐵𝐶
<C 
̅̅̅̅ 
𝐴𝐶
ΔABC is congruent
to ΔDEF because
___________________________
Therefore,
__________________________.
___________________________
.
Congruent Triangles are triangles with ______________
corresponding angles and _______________ _____________
sides.
Summary of Corresponding Parts
Example 1) If given the following picture, what congruence statement (conclusion) can be made? Justify your
answer.
Example 2) Mark the following diagram based on the given information.
Given: FE//AB,̅̅̅̅̅
𝐹𝐸 ≅ ̅̅̅̅
𝐴𝐵 , ̅̅̅̅̅
𝐹𝐶 ≅ ̅̅̅̅̅
𝐶𝐵 , ̅̅̅̅̅
𝐸𝐶 ≅ ̅̅̅̅
𝐴𝐶
Prove: ΔABC ≅ ΔEFC
Statements
1. FE//AB,̅̅̅̅̅
𝐹𝐸 ≅ ̅̅̅̅
𝐴𝐵 , ̅̅̅̅̅
𝐹𝐶 ≅ ̅̅̅̅̅
𝐶𝐵 ,
̅̅̅̅̅
̅̅̅̅
𝐸𝐶 ≅ 𝐴𝐶
Reasons
1.
Let’s work in reverse:
Example 1)
Given: ΔABC ≅ ΔXYZ
̅̅̅̅?
How might we prove that ̅̅̅̅
AC≅ XZ
(Hint: what property to do we need to show congruence?)
PRACTICE!
2. Identify all pairs of congruent corresponding parts in the triangles below. Then, write a congruence statement
for the triangles.
Corresponding angles:
Corresponding sides:
What do you know about these triangles. Explain.
3. Given: ΔABC ≅ ΔXYZ
Prove <ACB ≅ <XZY.
Statements
Reasons
4) In exercises 5-8, ΔXYZ ≅ ΔMNL. Complete each statement and explain how you got each answer!
a. m<Y = _____________________________
___________________________________
b. m<M = _____________________________
___________________________________
c. m<Z = _____________________________
___________________________________
d. XY = _______________________________________________________________
5) CD bisects AB at P. Prove AP = PB.
6) Given: E is the midpoint of AB.
Prove: ̅̅̅̅
𝐴𝐸 = ̅̅̅̅
𝐸𝐵
7) Find m<1:
e.
b.
8) Consider the following marking.
What was the given?
9) Given: Isosceles triangle ∆𝐴𝐵𝐶 with altitude AB.
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷.
Statement
Reason
1.
1. Given
2. AB ≅ BC
2.
3. BD ≅ BD
3.
4. BD is a perpendicular bisector and an
angle bisector.
4. Altitudes coincide with perpendicular
bisectors and angle bisectors in
_____________ triangles.
5. AD ≅ DC
5.
6. BD ⊥ AC
6.
7. ∠BDA and ∠ BDC are right angles
7.
8.
8. All right angles are congruent.
9. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷
9.
10. ∠A ≅ ∠C
10.
11. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷.
12. All corresponding angles are congruent
and all corresponding sides are
congruent.
Name_________________________________
Geometry Period _____
5-3 HW
Date______
1. List all pairs of congruent corresponding parts. Then, write a congruence statement for the triangles.
2. Find x and y in the following traingles. Explain how you got your answers!
3. Given: CD bisects AB.
Prove: AE = EB
4. Given: ̅̅̅̅
𝑨𝑫 is a perpendicular bisector and ̅̅̅̅
𝑨𝑫 is an angle
bisector. ΔABC is isosceles.
Prove that triangle ADC is congruent to triangle ADB.