Math by First Principles
Level 2 · Intermediate ← Back to lesson

Isosceles and Equilateral Triangles: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered
Question 1 of 12
  1. 1

    Triangle ABC has these side lengths:

    AB=AC.AB=2x+7.AC=5x−8.\begin{aligned} AB&=AC. \\ AB&=2x+7. \\ AC&=5x-8. \end{aligned}

    What is x?

    Keep the equal-side expressions togetherTriangle ABC has matching ticks on AB and AC. AB is labeled 2x plus 7 and AC is labeled 5x minus 8. No solved value is shown.ABC2x + 75x − 8Not to scale
  2. 2

    The base angles of an isosceles triangle have these measures:

    One base angle=(3x+8)∘.Other base angle=(5x−16)∘.\begin{aligned} \text{One base angle}&=(3x+8)^\circ. \\ \text{Other base angle}&=(5x-16)^\circ. \end{aligned}

    What is the vertex angle?

    Compare the base-angle expressionsAn isosceles triangle has matching side ticks and the two base angles labeled 3x plus 8 degrees and 5x minus 16 degrees. No solved angle is shown.ABC(3x + 8)°(5x − 16)°Not to scale
  3. 3

    An isosceles triangle has these angle measures:

    Vertex angle=(4x+12)∘.Each base angle=(x+18)∘.\begin{aligned} \text{Vertex angle}&=(4x+12)^\circ. \\ \text{Each base angle}&=(x+18)^\circ. \end{aligned}

    What is x?

    Count the two base anglesAn isosceles triangle has matching side ticks. Its vertex angle is labeled 4x plus 12 degrees and each base angle x plus 18 degrees.ABC(4x + 12)°(x + 18)°(x + 18)°Not to scale
  4. 4

    Triangle PQR has the equal sides

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

    Its base QR has length 29 centimeters. The internal angle bisector from P meets QR at M. What is QM?

    Draw the given vertex bisectorTriangle PQR has matching ticks on PQ and PR. PM bisects its vertex angle and meets QR at M. QR is labeled 29 centimeters. There are no midpoint ticks or solved half lengths.PQRMQR: 29 cmNot to scale
  5. 5

    A side of an isosceles triangle is extended at a base vertex. The exterior angle there is 128 degrees. What is the vertex angle between the equal sides?

    Locate the exterior angleAn isosceles triangle has matching side ticks. Its base is extended past the right base vertex, where the exterior angle is labeled 128 degrees. Interior angles are not labeled.ABC128°Not to scale
  6. 6

    An isosceles triangular sign has perimeter 72 centimeters and base 18 centimeters. What is the length of each equal side?

    Separate the base from the matching sidesAn isosceles triangular sign has matching side ticks and its base labeled 18 centimeters. The given perimeter is 72 centimeters. No equal-side length is supplied.ABCbase: 18 cmperimeter: 72 cmNot to scale
  7. 7

    Two side lengths of an equilateral triangle, in centimeters, are

    One side=7x−4.Another side=5x+12.\begin{aligned} \text{One side}&=7x-4. \\ \text{Another side}&=5x+12. \end{aligned}

    What is the length of the third side?

    Use the three equal sidesAn equilateral triangle has matching ticks on all sides. Two sides are labeled 7x minus 4 and 5x plus 12 centimeters. The third side and x are not evaluated.ABC7x − 45x + 12Not to scale
  8. 8

    Triangle LMN has the equal sides

    LM=LN.\begin{aligned} LM&=LN. \end{aligned}

    The internal bisector from L divides its vertex angle into two 31-degree angles. What is each base angle?

    Keep both given vertex partsTriangle LMN has matching ticks on LM and LN. Its vertex bisector divides angle L into two parts, each labeled 31 degrees. No base angle is labeled.LMN31°31°Not to scale
  9. 9

    Triangle ABC has the equal sides

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

    The altitude from A meets BC at D, with

    BD=2x+1.DC=x+8.\begin{aligned} BD&=2x+1. \\ DC&=x+8. \end{aligned}

    What is BC?

    Use the stated altitudeTriangle ABC has matching ticks on AB and AC. AD is perpendicular to BC, marked by a square at D. BD is labeled 2x plus 1 and DC x plus 8. No midpoint ticks are drawn.ABCD2x + 1x + 8Not to scale
  10. 10

    Which angle list can describe an obtuse isosceles triangle?

  11. 11

    Two equilateral triangles have equal angle measures. First triangle: each side is 10 centimeters.

    Second triangle: each side is 16 centimeters.

    Are they congruent?

    Compare the specified equilateral sidesTwo schematic equilateral triangles are compared vertically. The blue solid triangle has side length 10 centimeters. The red dashed triangle has side length 16 centimeters. Only the given side lengths are marked.ABCeach side: 10 cmDEFeach side: 16 cmNot to scale
  12. 12

    In an equilateral triangle, how many medians are also altitudes and internal angle bisectors?

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