Geometric Construction — Practice
With only a compass and a straightedge you can copy, bisect and build exactly — and the angle facts tell you when a figure is even possible.
What these questions cover
These questions use 14 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Complement — Everything except the event. P(not A) = 1 - P(A).
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Compass — Tools for drawing exact circles and lines — no measuring allowed.
- Complementary — Two angles summing to 90° / to 180°.
- Supplementary — Two angles summing to 90° / to 180°.
- Vertical angles — The opposite pair formed by two crossing lines — always equal.
- Base — The number being multiplied. In 53 the base is 5.
- Arc — Part of the edge; a "pizza slice" bounded by two radii.
- Transversal — A line that crosses two or more other lines.
- Co-interior angles — Between the parallels, same side. They add to 180°.
- Exterior angle — The angle formed by extending one side of a triangle.
- Parallel — Same slope, different intercept.
- Segment — Two endpoints and everything between them. Written AB.
- Limit — The value a function approaches as the input approaches some number.
Problem 1
Two lines cross. One angle measures 52°. Find its vertical angle.
Worked answer
52
52°
Problem 2
Find the supplement of 52°.
Worked answer
128
180 - 52 = 128°
Problem 3
Find the complement of 35°.
Worked answer
55
90 - 35 = 55°
Problem 4
A triangle has angles 50° and 60°. Find the third angle.
Worked answer
70
180 - 50 - 60 = 70°
Problem 5
Three angles around a point measure 120°, 150° and x. Find x.
Worked answer
90
360 - 120 - 150 = 90°
Problem 6
Can segments of length 5, 7 and 13 form a triangle?
- Yes
- No
Worked answer
No
5 + 7 = 12
12 < 13
The two short sides cannot reach across, so no triangle exists.
Problem 7
Can segments of length 4, 9 and 11 form a triangle?
- Yes
- No
Worked answer
Yes
4 + 9 = 13 > 11 ✓
A triangle is possible.
Problem 8
Two parallel lines are cut by a transversal. One angle is 73°. Find the co-interior angle on the same side.
Worked answer
107
180 - 73 = 107°
Problem 9
A quadrilateral has angles 85°, 95° and 110°. Find the fourth.
Worked answer
70
360 - 85 - 95 - 110 = 70°
Problem 10
A triangle has two angles of 40° each. Find the third.
Worked answer
100
180 - 40 - 40 = 100°
Problem 11
When bisecting an angle with a compass, why must the compass width stay the same for the two final arcs?
- So the drawing looks neat
- So the crossing point is the same distance from both sides
- So the arcs fit on the page
- It does not actually matter
Worked answer
So the crossing point is the same distance from both sides
Equal radii put the crossing point the same distance from each side of the angle.
A point equidistant from both sides lies exactly on the bisector — that is what makes the construction a proof, not a guess.
Problem 12
A triangle has interior angles of 60° and 70°. Find the exterior angle at the third vertex.
Worked answer
130
Third interior angle: 180 - 60 - 70 = 50°
Exterior angle: 180 - 50 = 130°
Or directly: 60 + 70 = 130° ✓
Problem 13
Two sides of a triangle are 7 and 11. How many whole-number values are possible for the third side?
Worked answer
13
11 - 7 = 4 and 11 + 7 = 18
So 4 < x < 18
Whole numbers from 5 to 17
17 - 5 + 1 = 13 values
Problem 14
Find one interior angle of a regular hexagon.
Worked answer
120
(6 - 2) × 180 = 720°
Regular, so all six are equal
720 ÷ 6 = 120°
Problem 15
An isosceles triangle has an apex angle of 40°. Find each base angle.
Worked answer
70
180 - 40 = 140° for the two base angles
140 ÷ 2 = 70° each
Problem 16
If you are given only the three angle measures of a triangle, how many different triangles fit?
- Exactly one
- Exactly two
- None
- Infinitely many
Worked answer
Infinitely many
Angles fix the shape but not the size.
Every triangle with those angles is similar to the others, and there is no limit on how large you make it.
So infinitely many — this is why three angles do not determine a unique triangle, while three sides do.