Math by First Principles
Level 3 · Challenge ← Back to lesson

Isosceles and Equilateral Triangles: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered
Question 1 of 12
  1. 1

    Triangle ABC has the equal sides

    AB=AC.\begin{aligned} AB&=AC. \end{aligned}

    Angle BAC measures 100 degrees. D lies inside BC, with

    AD=BD.\begin{aligned} AD&=BD. \end{aligned}

    What is angle DAC?

    Two equal-side pairs in a divided triangleTriangle ABC has AB equal to AC, D inside BC, and AD equal to BD. The whole vertex angle BAC is given as 100 degrees. Single side ticks mark AB and AC; double side ticks mark AD and BD. No derived angle is labeled.ABCDGiven angle BAC: 100°
  2. 2

    Triangle ABC has the equal sides

    AB=AC.\begin{aligned} AB&=AC. \end{aligned}

    Angle BAC measures 36 degrees. D is inside AC, with

    BD=BC.\begin{aligned} BD&=BC. \end{aligned}

    What is angle ABD?

    A second isosceles triangle inside the firstTriangle ABC has the given side equality AB equals AC stated in a label. D lies inside AC at the recorded position, and BD and BC have matching double ticks. Only the given 36-degree angle at A is labeled. No tick divides AC into subsegments.ABCD36°Given: AB = AC
  3. 3

    An isosceles triangle has these angle measures:

    Each base angle=(x−1)∘.Vertex angle=(4x+2)∘.\begin{aligned} \text{Each base angle}&=(x-1)^\circ. \\ \text{Vertex angle}&=(4x+2)^\circ. \end{aligned}

    What is the vertex angle?

    Relate the repeated base anglesAn isosceles triangle has matching side ticks. Each base angle is labeled x minus 1 degrees and the vertex angle 4x plus 2 degrees. No numerical answer is shown.ABC(4x + 2)°(x − 1)°(x − 1)°Not to scale
  4. 4

    In triangle PQR, the equal sides are

    PQ=PR.\begin{aligned} PQ&=PR. \end{aligned}

    M is the midpoint of QR. A proof compares triangles PQM and PRM. Which third fact completes the SSS justification?

    Use the marked midpointTriangle PQR has matching ticks on PQ and PR. M is on QR, whose two halves have matching double ticks. PM is drawn without right-angle or angle-bisector marks.PQRMNot to scale
  5. 5

    In triangle ABC, angles B and C are congruent. The conclusion to be proved is

    AB=AC.\begin{aligned} AB&=AC. \end{aligned}

    Compare triangle BCA with triangle CBA. Which fact provides the included side for ASA without assuming that conclusion?

    Use the given equal-angle pairTriangle ABC has matching arcs at B and C. No side equalities are marked.ABCNot to scale
  6. 6

    Which argument proves that an equilateral triangle is equiangular?

    Begin with the equilateral givensTriangle ABC has a matching tick on each side. No equal-angle arcs or angle measures are drawn.ABCNot to scale
  7. 7

    Triangle ABC has the equal sides

    AB=AC.\begin{aligned} AB&=AC. \end{aligned}

    The altitude from A meets BC at D. Which conclusion must hold?

    Separate altitude givens from conclusionsTriangle ABC has matching ticks on AB and AC. AD meets BC at a marked right angle. No midpoint ticks or equal vertex-angle arcs are drawn.ABCDNot to scale
  8. 8

    Triangle XYZ has congruent angles at X and Y. Through which vertex is a reflection axis guaranteed to pass?

    Locate the given equal anglesTriangle XYZ has matching arcs at X and Y. No equal-side ticks or symmetry axis is drawn.ZXYNot to scale
  9. 9

    Which condition guarantees that a nondegenerate isosceles triangle has three reflection axes?

  10. 10

    Two isosceles triangles each have a vertex angle of 104 degrees. First triangle: both equal sides are 13 centimeters.

    Second triangle: both equal sides are 17 centimeters.

    Which statement is justified?

    Compare the specified equal sidesTwo schematic isosceles triangles are compared vertically. Each vertex angle is given as 104 degrees. Blue solid has matching sides of 13 centimeters; red dashed has matching sides of 17 centimeters. No base angle is labeled.ABCequal sides: 13 cm104°DEFequal sides: 17 cm104°Not to scale
  11. 11

    The median from the vertex between the equal sides divides an isosceles triangle into congruent halves. The two angles at its foot are both t degrees and form a linear pair. Which equation establishes that this median is also an altitude?

    Use the equal adjacent anglesAn isosceles triangle has a median from its common equal-side vertex to the midpoint of the base. Matching base-half ticks mark the midpoint. Each adjacent foot angle is labeled t degrees, with no right-angle square or value of t.ABCDt°t°Not to scale
  12. 12

    In triangle ABC, D lies inside BC, AD is perpendicular to BC, angle BAD is 30 degrees, and angle DAC is 40 degrees. What does this show about a claim that every altitude halves the angle at its starting vertex?

    Compare the given vertex partsA schematic triangle ABC has AD perpendicular to BC at D. Angle BAD is labeled 30 degrees and angle DAC 40 degrees. No equal-side or equal-angle marks are drawn.ABCD30°40°Not to scale
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