Triangle Congruence Proofs — Challenge
Six parts make a triangle, but you never need all six. Three of the right kind are enough.
What these problems 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
Two sides of one triangle and the angle between them equal the matching parts of another. Which test proves the triangles congruent?
- SAS
- SSA
- AAA
- ASS
- AA
Worked answer
SAS
Two sides with the angle between them is side-angle-side, written SAS.
Fixing two sides and the angle they enclose fixes the third side as well, so the triangle is determined completely.
SSA is not a valid test: the same two sides and a non-included angle can give two different triangles.
Problem 2
What is true of the base angles of an isosceles triangle?
- They are equal
- They are supplementary
- They are complementary
- They are right angles
- They are unequal
Worked answer
They are equal
An isosceles triangle has two equal sides.
Reflecting it in the line through the apex and the midpoint of the base maps the triangle onto itself and swaps the base angles.
Since reflection preserves angles, the two base angles are equal.
Problem 3
Two angles of a triangle are equal. What kind of triangle is it?
- Scalene
- Isosceles
- Equilateral
- Right-angled
- Obtuse
Worked answer
Isosceles
The sides opposite equal angles in a triangle are themselves equal.
Two equal angles therefore force two equal sides, which is the definition of isosceles.
It need not be equilateral: that would require all three angles equal.
Problem 4
How many degrees is each angle of an equilateral triangle?
- 30
- 45
- 60
- 90
- 120
Worked answer
60
All three sides are equal, so all three angles are equal.
The angles of any triangle add to 180 degrees.
So each angle is 1803 = 60 degrees.
Problem 5
An exterior angle of a triangle is equal to what?
- The sum of the two opposite interior angles
- The adjacent interior angle
- 90 degrees
- The largest interior angle
- 180 degrees
Worked answer
The sum of the two opposite interior angles
The exterior angle and its adjacent interior angle sit on a straight line, so they add to 180 degrees.
The three interior angles also add to 180 degrees.
Comparing the two statements shows the exterior angle equals the sum of the other two interior angles.
Problem 6
If all three sides of one triangle equal the three sides of another, what does that prove?
- Congruence
- Similarity only
- Nothing
- Equal area only
- Equal perimeter only
Worked answer
Congruence
Three fixed side lengths determine a triangle completely: there is no freedom left to change its angles.
So the two triangles must be identical in shape and size, which is congruence. This is the SSS test.
Congruence implies equal area and perimeter, so those weaker statements follow rather than being the best answer.
Problem 7
Which of these does not prove that two triangles are congruent?
- SSS
- SAS
- ASA
- AAA
- RHS
Worked answer
AAA
Each valid test pins down at least one length, which fixes the size of the triangle.
AAA gives only the three angles, which fixes the shape but not the size.
Two triangles with equal angles may be of very different sizes, so AAA proves similarity rather than congruence.
Problem 8
In triangle ABC, AB=AC. Which two angles are equal?
- B and C
- A and B
- A and C
- All three
- None of them
Worked answer
B and C
The equal sides are AB and AC, which both meet at vertex A.
Each angle is opposite the side it does not touch: angle B faces AC and angle C faces AB.
Since those two sides are equal, angles B and C are equal. These are the base angles.