Triangle Congruence Proofs — Drills
Triangle Congruence Proofs has 8 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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)!.
Choosing the method
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
You are given three pairs of congruent sides. Name the congruence shortcut.
Worked answer
SSS
Three pairs of congruent sides.
That is SSS.
Problem 2
You are given two angles and the side between them. Name the congruence shortcut.
Worked answer
ASA
Two angles and the side between them.
That is ASA.
Reflexive
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two triangles share side BD. What reason gives BD ≅ BD?
Worked answer
reflexive property
BD ≅ BD
That is the reflexive property — and it is the free third part in most shared-side proofs.
Problem 2
Two triangles share side AC. What reason gives AC ≅ AC?
Worked answer
reflexive property
AC ≅ AC
That is the reflexive property — and it is the free third part in most shared-side proofs.
Cpctc
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Given △ ABC ≅ △ DEF, which part corresponds to AC?
Worked answer
\overline{DF
The order of the letters is the correspondence.
So AC pairs with DF, by CPCTC.
Problem 2
Given △ ABC ≅ △ DEF, which part corresponds to BC?
Worked answer
\overline{EF
The order of the letters is the correspondence.
So BC pairs with EF, by CPCTC.
Isosceles
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
In △ ABC, AB ≅ AC and m∠ B = 54°. Find m∠ C.
Worked answer
54
Equal sides give equal opposite angles.
m∠ C = 54°
Problem 2
In △ ABC, AB ≅ AC and m∠ B = 45°. Find m∠ C.
Worked answer
45
Equal sides give equal opposite angles.
m∠ C = 45°
Hl
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
HL may be used on
- right triangles only
- any triangle
- obtuse triangles only
- isosceles triangles only
Worked answer
right triangles only
H is for hypotenuse, which only exists in a right triangle.
So HL applies to right triangles only.
Vertical
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two triangles meet at a point so a pair of angles are vertical. Are those angles congruent? Answer yes or no.
Worked answer
yes
Vertical angles are always congruent.
No parallel lines are required, which is why this is such a common free third part.
Solving
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
△ ABC ≅ △ DEF with AB = 5x + 2 and DE = 4x + 11. Find AB.
Worked answer
47
5x + 2 = 4x + 11
x = 9
AB = 5(9) + 2 = 47
Problem 2
△ ABC ≅ △ DEF with AB = 5x + 17 and DE = 2x + 56. Find AB.
Worked answer
82
5x + 17 = 2x + 56
x = 13
AB = 5(13) + 17 = 82
Proof
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Given ∠ A ≅ ∠ C and ∠ ADB ≅ ∠ CDB, with BD shared. Name the shortcut proving the two triangles congruent.
Worked answer
AAS
The shared side is congruent to itself.
That gives two angles and a non-included side — AAS.
Problem 2
Given AB ≅ CB and AD ≅ CD, with BD shared. Name the shortcut proving the two triangles congruent.
Worked answer
SSS
The shared side is congruent to itself.
That gives all three sides — SSS.