A triangle has angles 75° and 67°. Find the third angle.
Answer: 38
Reasoning: Triangle angles sum to 180°.
Reasoning: Triangle angles sum to 180°.
Unit 03 · Lesson 07 · Triangle Congruence & Proof
Perpendicular Bisectors, Angle Bisectors & Centers · Teacher Key
Construct and compare circumcenter, incenter, centroid, and orthocenter locations.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to construct and compare circumcenter, incenter, centroid, and orthocenter locations.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Identify sufficient evidence for triangle congruence and organize theorem-based proofs about corresponding parts and centers.
m\angle A+m\angle B+m\angle C=180^\circa+b>cSSS,\ SAS,\ ASA,\ AAS,\ HL
Worked reasoning
Process: Match vertices in a consistent correspondence, mark the known evidence, select the valid congruence theorem, then use CPCTC only after congruence is established.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: AAA proves similarity, not congruence; SSA is not a general congruence theorem.
Khaemenes Academy · Open-Age Geometry
Perpendicular Bisectors, Angle Bisectors & Centers
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Two sides are 8 and 8. State the integer range for the third side x.
Answer: 0<x<16
Reasoning: Use the triangle inequality and the difference inequality.
Reasoning: Use the triangle inequality and the difference inequality.
Which congruence theorem uses two sides and the included angle?
Answer: SAS
Reasoning: The angle must be between the two known sides.
Reasoning: The angle must be between the two known sides.
After proving ΔABC≅ΔDEF, which angle corresponds to ∠B?
Answer: ∠E
Reasoning: The order of vertices determines correspondence.
Reasoning: The order of vertices determines correspondence.
Where do the perpendicular bisectors of a triangle meet?
Answer: Circumcenter
Reasoning: It is equidistant from all three vertices.
Reasoning: It is equidistant from all three vertices.
Why does AAA not prove triangle congruence?
Answer: It fixes shape but not size
Reasoning: AAA establishes similarity.
Reasoning: AAA establishes similarity.