Unit 4: Triangle Congruence and Proof
The shortcuts that prove two triangles match without checking all six parts, why one shortcut secretly fails, and how to write the argument down.
10 topics · 35 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Congruent Triangles and CPCTC
- How it worksHow does naming triangles in matching vertex order tell you exactly which sides and angles correspond?
- Worked exampleFind the measure of angle F if triangle ABC is congruent to triangle DEF and angle C measures 72 degrees.
- Common mistakeWhy isn't it enough for two triangles to just have the same shape without the same size to be congruent?
- Going furtherWhat happens if you try to use CPCTC before the two triangles have actually been proven congruent?
Your turn1 of 2
Triangle ABC is congruent to triangle DEF, matched in that vertex order (A to D, B to E, C to F). Angle C measures 121 degrees. Find the measure of angle F.SSS Congruence
- How it worksHow does the side-side-side postulate prove two triangles congruent using only their three side lengths?
- Worked exampleShow that a triangle with sides 5, 12, 13 is congruent to a triangle with sides 13, 5, 12 using SSS.
- Going furtherWhat happens to an SSS congruence claim if only two of the three side lengths actually match?
- Going furtherWhat happens to SSS congruence when you're only given two sides and told nothing about the third?
Your turn1 of 2
A triangle has sides 9, 9, and 7. A second triangle has the same three side lengths, just listed in a different order. By SSS, the two triangles are congruent. Find their shared perimeter.SAS Congruence
- How it worksHow does the side-angle-side postulate use an included angle to prove two triangles congruent?
- Worked exampleShow that two triangles are congruent by SAS when both have sides of 6 and 9 with an included angle of 40 degrees.
- Common mistakeWhy does the angle in SAS have to sit between the two given sides instead of anywhere in the triangle?
Your turn1 of 2
A triangle has two sides of length 11 and 11, with an included angle of 159 degrees between them. Using the Law of Cosines, find the length of the third side.ASA and AAS Congruence
- How it worksHow does angle-side-angle use an included side between two known angles to prove congruence?
- How it worksHow does angle-angle-side prove congruence even when the known side isn't between the two angles?
- Worked exampleProve two triangles congruent by ASA given angles of 50 and 65 degrees and an included side of 8.
- Common mistakeWhy doesn't AAA, knowing all three angles, ever prove two triangles congruent?
Your turn1 of 2
Two triangles each have angles of 62 degrees and 59 degrees, with a side of length 19 included between them. Find the measure of the third angle of each triangle.Why SSA Doesn't Prove Congruence
- How it worksHow does swinging the side opposite the given angle sometimes land on two different possible triangles?
- Worked exampleFind both possible triangles when a side of 7, a side of 10, and a non-included angle of 35 degrees are given.
- Common mistakeWhy isn't side-side-angle a valid shortcut even though it lists three matching parts?
- Going furtherWhat has to be true about the given angle for SSA to accidentally produce only one triangle instead of two?
Your turn1 of 2
A triangle attempt has a side of length 18, a side of length 6, and a non-included angle of 57 degrees, opposite the side of length 18. Using the Law of Sines, find the sine of the angle opposite the side of length 6.HL Congruence for Right Triangles
- How it worksHow does the Pythagorean theorem secretly turn hypotenuse-leg into a hidden case of SSS?
- Worked exampleProve two right triangles congruent by HL given hypotenuses of 13 and legs of 5.
- Common mistakeWhy doesn't HL work on a triangle that isn't already known to have a right angle?
Your turn1 of 2
Two right triangles each have a hypotenuse of length 17 and one leg of length 7. By HL, the triangles are congruent. Find the length of the other leg.Two-Column Proofs
- How it worksHow does each line of a two-column proof have to follow logically from a previous line or a given fact?
- Worked exampleProve in two-column form that triangle ABC is congruent to triangle ADC given that AB equals AD and BC equals DC.
- Common mistakeWhy can't a two-column proof skip a step just because the conclusion looks obvious from the picture?
Your turn1 of 2
In a two-column proof, why can't a step be skipped just because the conclusion looks obvious from the picture?Paragraph Proofs
- How it worksHow does a paragraph proof still have to name every definition, postulate, and theorem it leans on?
- Worked exampleProve in paragraph form that the base angles of an isosceles triangle are congruent.
- Ties togetherHow does a paragraph proof and a two-column proof of the same fact use exactly the same logical steps in different clothing?
Your turn1 of 2
A two-column proof and a paragraph proof are written for the exact same geometric fact. How do their underlying logical arguments compare?Isosceles Triangle Theorems
- How it worksHow does drawing the angle bisector from the apex of an isosceles triangle split it into two congruent triangles?
- Worked exampleFind the base angles of an isosceles triangle whose vertex angle measures 44 degrees.
- Common mistakeWhy isn't every triangle with one pair of congruent angles automatically equilateral?
- Going furtherWhat happens to the isosceles triangle theorem when all three sides happen to be equal?
Your turn1 of 2
An isosceles triangle has a vertex (apex) angle measuring 121 degrees. Find the measure of each base angle.What a Diagram Does and Doesn't Tell You
- How it worksHow do you separate what a diagram explicitly states from what it only happens to look like?
- Common mistakeWhy can't you assume two segments are congruent just because they're drawn the same length on paper?
- Going furtherWhat happens to a proof's validity if it secretly relies on an angle looking like 90 degrees in the drawing?
Your turn1 of 2
A diagram shows two segments drawn at what appears to be the same length, but no tick marks or given statement say they are congruent. Do you need a stated or proven fact before using their congruence in a proof?