Points, Lines, Planes, and Angle Relationships: math practice, Grade 10 – download the PDF

1 Point, line, or plane? ★★★

For each object, say whether it is best modeled by a point, a line, or a plane.

1. A dot marking a city on a map
2. A perfectly straight road that continues forever in both directions
3. The flat surface of a calm lake, imagined to extend forever
4. The tip of a sharpened pencil
5. A huge sheet of glass with no edges

2 Adding two segments ★★★

Point \(Q\) is between \(P\) and \(R\). If \(PQ = 14.5\text{ cm}\) and \(QR = 9.3\text{ cm}\), find \(PR\).

3 Midpoint on a number line ★★★

A segment on a number line has endpoints at \(-6\) and \(10\). Find its length and its midpoint.

4 Complement and supplement ★★★

An angle measures \(37^\circ\). Find the measure of its complement and of its supplement.

5 Vertical angles ★★★

Two lines intersect. One of the four angles measures \(48^\circ\). Find the other three.

6 A linear pair ★★★

Two angles form a linear pair. One measures \(x^\circ\) and the other measures \((x + 30)^\circ\). Find both angles.

7 Angle addition ★★★

Ray \(OB\) lies inside the right angle \(\angle AOC\). If \(m\angle AOB = 28^\circ\), find \(m\angle BOC\).

8 Segment addition with algebra ★★★

Point \(M\) is between \(L\) and \(N\). \(LM = (3x + 2)\), \(MN = (2x + 7)\) and \(LN = 44\), all in centimeters. Find \(x\), then \(LM\) and \(MN\).

9 Midpoint and distance ★★★

Let \(A(-3, 8)\) and \(B(9, -2)\). Find the midpoint of \(\overline{AB}\) and the exact length \(AB\), then round the length to the nearest hundredth.

10 Find the other endpoint ★★★

The midpoint of \(\overline{AB}\) is \(M(4, -1)\) and one endpoint is \(A(7, 2)\). Find \(B\).

11 Bisector equation ★★★

Ray \(BD\) bisects \(\angle ABC\). \(m\angle ABD = (4x + 3)^\circ\) and \(m\angle DBC = (6x - 17)^\circ\). Find \(x\) and \(m\angle ABC\).

12 Supplement equals three complements ★★★

The supplement of an angle is exactly three times its complement. Find the angle.

13 Vertical angles with variables ★★★

Two lines intersect. Two vertical angles measure \((5x - 12)^\circ\) and \((3x + 20)^\circ\). Find \(x\), the measure of these angles, and the measure of each of the other two angles.

14 True or false? ★★★

Decide whether each statement is true or false. Justify with a reason or a counterexample.

1. If two angles are supplementary, they form a linear pair.
2. Vertical angles are always congruent.
3. Every segment has exactly one midpoint.

15 Intersecting lines ★★★

Two lines intersect, forming \(\angle 1\), \(\angle 2\), \(\angle 3\), and \(\angle 4\) in order around the point. \(\angle 1\) and \(\angle 2\) form a linear pair with \(m\angle 1 = (2x + 10)^\circ\) and \(m\angle 2 = (3x - 5)^\circ\).

Find \(x\) and the measures of all four angles.

16 A hiking map ★★★

On a trail map, one unit on each axis is 1 mile. The trailhead is at \(T(2, 3)\) and the cabin is at \(C(14, 12)\). The hikers plan to rest at the midpoint of the straight path from \(T\) to \(C\).

1. Find the coordinates of the rest stop.
2. Find the straight-line distance from the trailhead to the cabin in miles, and convert it to kilometers (1 mile \(\approx\) 1.609 km).

17 Four points on a fence ★★★

Points \(A\), \(B\), \(C\), \(D\) lie in this order along a straight fence. \(AB = 2x\), \(BC = (x + 3)\), \(CD = (3x - 1)\) and \(AD = 50\) feet.

1. Find \(x\) and the length of each section.
2. The midpoint of \(\overline{AD}\) lies on which section? How far is it from \(B\)?

18 Three adjacent angles ★★★

Rays \(OB\) and \(OC\) lie inside \(\angle AOD\), which measures \(150^\circ\). \(m\angle AOB = x^\circ\), \(m\angle BOC = (2x + 10)^\circ\) and \(m\angle COD = (3x - 4)^\circ\).

1. Find \(x\) and each angle.
2. Does the bisector of \(\angle AOD\) lie between \(OB\) and \(OC\)? Explain.

19 Proof: vertical angles ★★★

Two lines intersect and form \(\angle 1\), \(\angle 2\), \(\angle 3\) in order (so \(\angle 1\) and \(\angle 3\) are vertical angles). Complete a two-column proof that \(m\angle 1 = m\angle 3\).

20 Bisectors of a linear pair ★★★

Two angles form a linear pair. A ray bisects each of them. Show that the two bisectors are perpendicular. Then check with a linear pair of \(50^\circ\) and \(130^\circ\).

21 Endpoint and length ★★★

The midpoint of \(\overline{PQ}\) is \(M(-1.5, 4)\) and \(P(-5, 7)\). Find \(Q\) and the length \(PQ\) to the nearest hundredth.

22 Why does the construction work? ★★★

Describe how to bisect \(\angle ABC\) with a compass and straightedge, then explain why the ray you draw really splits the angle into two equal angles.