High School Geometry Test Questions on Equilateral Triangle Proofs

📚 Quick Study Guide

• 📐 Definition: An equilateral triangle is a triangle in which all three sides are equal in length.
• 📏 Properties:
  • All three angles are congruent (equal).
  • Each angle measures 60 degrees.
  • It is also an equiangular triangle.
• 📐 Theorems & Postulates:
  • Side-Side-Side (SSS) Congruence Postulate: If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.
  • Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle, then the two triangles are congruent.
  • Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.
  • Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
  • Converse of Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
• 💡 Key Proof Strategies:
  • Start with the given information.
  • Use definitions and theorems to justify each step.
  • Look for congruent sides and angles.
  • Use congruence postulates (SSS, ASA, SAS) to prove triangle congruence.

✍️ Practice Quiz

1. Which of the following statements is always true for an equilateral triangle?
   1. A. All sides are of different lengths.
   2. B. All angles are 90 degrees.
   3. C. All sides are congruent.
   4. D. Only two angles are congruent.
2. If $\triangle ABC$ is equilateral, what is the measure of $\angle A$?
   1. A. 30 degrees
   2. B. 45 degrees
   3. C. 60 degrees
   4. D. 90 degrees
3. Given $\triangle PQR$ is equilateral with $PQ = 5$, what is the length of $QR$?
   1. A. 2.5
   2. B. 5
   3. C. 7.5
   4. D. 10
4. Which postulate can be used to prove that two equilateral triangles are congruent if all their sides are equal?
   1. A. ASA
   2. B. SAS
   3. C. SSS
   4. D. AAS
5. If $\triangle XYZ$ is equilateral and $XY = 2x + 3$ and $YZ = x + 7$, find the value of $x$.
   1. A. 2
   2. B. 3
   3. C. 4
   4. D. 5
6. In equilateral $\triangle DEF$, if $\angle D = (3y - 15)$ degrees, what is the value of $y$?
   1. A. 15
   2. B. 20
   3. C. 25
   4. D. 30
7. If two equilateral triangles share a common side, what is the measure of the angle formed by the non-shared sides at a vertex of the shared side?
   1. A. 60 degrees
   2. B. 90 degrees
   3. C. 120 degrees
   4. D. 180 degrees