Triangle Congruence Proofs — Practice
Six parts make a triangle, but you never need all six. Three of the right kind are enough.
What these questions cover
These questions use 15 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Corresponding sides — Sides in matching positions in two similar figures.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Vertical angles — The opposite pair formed by two crossing lines — always equal.
- Base — The number being multiplied. In 53 the base is 5.
- Hypotenuse — The side opposite the right angle — always the longest.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Corresponding parts — The pieces that match up: A with D, side AB with side DE, and so on.
- Parallel — Same slope, different intercept.
- Prove — The statement you must arrive at. Always the last line.
- Included angle — The angle between two named sides. The A in SAS must be included.
- Included side — The side between two named angles. The S in ASA must be included.
- CPCTC — Corresponding Parts of Congruent Triangles are Congruent — used after proving congruence.
- Reflexive property — Anything is congruent to itself. A shared side is congruent to itself.
- Ambiguous case — Two sides and a non-included angle, which can describe two different triangles.
- Combination — A selection where order does not matter. C(n,r) = n!r!(n-r)!.
Problem 1
Given three pairs of congruent sides, which shortcut applies?
- SSS
- SAS
- ASA
- AAS
Worked answer
SSS
Three sides, no angles.
That is SSS.
Problem 2
Two angles and the side between them are congruent. Which shortcut?
- ASA
- AAS
- SAS
- SSA
Worked answer
ASA
The side sits between the two angles.
That is ASA.
Problem 3
Which combination does NOT prove congruence?
- SSA
- SSS
- ASA
- AAS
Worked answer
SSA
SSA can give two different triangles.
It proves nothing.
Problem 4
Two triangles share side BD. What reason gives BD ≅ BD?
Worked answer
reflexive property
BD ≅ BD by the reflexive property.
Problem 5
After proving △ ABC ≅ △ DEF, what reason lets you write ∠ B ≅ ∠ E?
Worked answer
CPCTC
B pairs with E in the correspondence.
So ∠ B ≅ ∠ E by CPCTC.
Problem 6
AB ≅ DE, ∠ B ≅ ∠ E, BC ≅ EF. Name the shortcut.
Worked answer
SAS
∠ B is formed by AB and BC.
Included angle, so SAS.
Problem 7
∠ A ≅ ∠ D, ∠ B ≅ ∠ E, BC ≅ EF. Name the shortcut.
Worked answer
AAS
BC is not between ∠ A and ∠ B.
Two angles and a non-included side: AAS.
Problem 8
In △ ABC, AB ≅ AC and m∠ B = 52°. Find m∠ C.
Worked answer
52
Equal sides give equal opposite angles.
m∠ C = 52°
Problem 9
In △ ABC, AB ≅ AC and m∠ A = 40°. Find m∠ B.
Worked answer
70
180 - 40 = 140
140 ÷ 2 = 70°
Problem 10
HL may be used on
- right triangles only
- any triangle
- isosceles triangles only
- obtuse triangles only
Worked answer
right triangles only
Hypotenuse — so there must be a right angle.
HL is right triangles only.
Problem 11
Two triangles meet at a point so that a pair of angles are vertical. Are those angles congruent? Answer yes or no.
Worked answer
yes
No parallel lines needed.
Vertical angles are congruent by theorem.
Problem 12
Two triangles have all three angles congruent. Are they congruent? Answer yes or no.
Worked answer
no
AAA gives similar triangles.
They could be any size, so not necessarily congruent.
Problem 13
△ ABC ≅ △ DEF with AB = 3x + 1 and DE = 2x + 7. Find AB.
Worked answer
19
3x + 1 = 2x + 7, so x = 6
AB = 3(6) + 1 = 19
Problem 14
An isosceles triangle has a base angle of (3x)° and vertex angle (2x + 20)°. Find x.
Worked answer
20
3x + 3x + 2x + 20 = 180
8x = 160
x = 20
Problem 15
Given AB ≅ CB and ∠ ABD ≅ ∠ CBD, with BD shared. Name the shortcut proving △ ABD ≅ △ CBD.
Worked answer
SAS
Side AB, included angle ∠ ABD, side BD (reflexive).
That is SAS.
Problem 16
From △ ABD ≅ △ CBD, which side corresponds to AD?
Worked answer
CD
AD pairs with CD.
AD ≅ CD by CPCTC.