Geometry Points, Lines, Segments and Angles Worksheet Solutions (PDF)

Geometry math worksheet solutions: Points, Lines, Segments and Angles – Maths PDF

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This basics practice solutions for Geometry covers points, lines, segments and angles (Common Core G-CO.A.1), with a free PDF to print.

Use this answer key to check the 20 problems on segments and angles. Every solution shows the formula or theorem used, each step of the work, and the final answer in bold. If an answer differs from yours, compare the steps line by line to find where the work changed.

Geometry · Common Core State Standards: G-CO.A.1

Basics practice solutions: exercises 1 to 10

Exercise 1 – Length on a number line

Length is the absolute value of the difference of the coordinates.

\(AB = |9 – (-3)| = |12| = 12\)

AB = 12

Exercise 2 – Midpoint on a number line

\(\frac{4 + 18}{2} = \frac{22}{2} = 11\)

Exercise 3 – Segment addition

AB + BC = AC.

B is between A and C, so use the Segment Addition Postulate.

\(AC = AB + BC = 14 + 9 = 23\)

AC = 23

Exercise 4 – Segment with a variable

\((3x – 2) + (x + 8) = 34\)

\(4x + 6 = 34\)

\(4x = 28\)

\(x = 7\)

\(AB = 3(7) – 2 = 19\) and \(BC = 7 + 8 = 15\). Check: 19 + 15 = 34.

x = 7, AB = 19, BC = 15

Exercise 5 – Midpoint with a variable

A midpoint makes the two parts equal.

\(2x + 3 = 5x – 9\)

\(12 = 3x\)

\(x = 4\)

\(AM = 2(4) + 3 = 11\), so \(AB = 2 \cdot 11 = 22\).

x = 4, AB = 22

Exercise 6 – Angle addition

35° + 48° = 83°.

Ray BD is inside the angle, so use the Angle Addition Postulate.

\(m\angle ABC = 35^{\circ} + 48^{\circ} = 83^{\circ}\)

83°

Exercise 7 – Angle bisector

A bisector splits the angle into two equal parts.

\(m\angle ABD = 76^{\circ} \div 2 = 38^{\circ}\)

38°

Exercise 8 – Bisector with a variable

The two angles are equal.

\(4x + 6 = 7x – 24\)

\(30 = 3x\)

\(x = 10\)

\(4(10) + 6 = 46\), so each half is \(46^{\circ}\).

\(m\angle ABC = 2 \cdot 46^{\circ} = 92^{\circ}\)

x = 10, m∠ABC = 92°

Exercise 9 – Complement and supplement

\(90^{\circ} – 37^{\circ} = 53^{\circ}\) (complement)

\(180^{\circ} – 37^{\circ} = 143^{\circ}\) (supplement)

Complement 53°, supplement 143°

Exercise 10 – Linear pair

A linear pair is supplementary.

\((3x + 10) + (5x – 6) = 180\)

\(8x + 4 = 180\)

\(8x = 176\)

\(x = 22\)

\(3(22) + 10 = 76\) and \(5(22) – 6 = 104\). Check: 76 + 104 = 180.

x = 22; 76° and 104°

Exercises 11 and up

Exercise 11 – Midpoint in the plane

The midpoint is M(3, 2).

\(\left( \frac{-2 + 8}{2}, \frac{5 + (-1)}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2)\)

M(3, 2)

Exercise 12 – Find an endpoint

\(\frac{-3 + x}{2} = 1\) gives \(x = 5\).

\(\frac{4 + y}{2} = -2\) gives \(y = -8\).

B(5, -8)

Exercise 13 – Distance in the plane

\(\sqrt{(7 – 1)^{2} + (10 – 2)^{2}} = \sqrt{36 + 64} = \sqrt{100} = 10\)

10 units

Exercise 14 – Perpendicular lines

Perpendicular lines form right angles.

\(2x + 10 = 90\)

\(2x = 80\)

\(x = 40\)

x = 40

Exercise 15 – Points on a circle

\(OP = \sqrt{6^{2} + 8^{2}} = \sqrt{100} = 10\)

\(OQ = \sqrt{(-10)^{2} + 0^{2}} = 10\)

Q is also 10 units from the center, so it lies on the circle.

Radius 10; yes, Q is on the circle

Exercise 16 – Trail rest stop

\(\frac{3.2 + 11.6}{2} = \frac{14.8}{2} = 7.4\)

\(11.6 – 3.2 = 8.4\), and half of 8.4 is 4.2.

Mile 7.4, which is 4.2 miles from each end

Exercise 17 – Fence line

\(BC = x\) and \(AB = 2x\).

\(2x + x = 45\)

\(3x = 45\)

\(x = 15\)

BC = 15 ft and AB = 30 ft

Exercise 18 – Garden path

The bisector makes each half \(120^{\circ} \div 2 = 60^{\circ}\).

\(5x + 10 = 60\)

\(5x = 50\)

\(x = 10\)

x = 10

Exercise 19 – Angle between clock hands

At 4:00 the hands are 4 hours apart, and each hour is \(360^{\circ} \div 12 = 30^{\circ}\), so \(4 \cdot 30^{\circ} = 120^{\circ}\).

At 2:30 the minute hand is on the 6, which is \(180^{\circ}\) from the 12. The hour hand moved halfway from 2 to 3: \(2.5 \cdot 30^{\circ} = 75^{\circ}\).

\(180^{\circ} – 75^{\circ} = 105^{\circ}\)

2:30 gives 105°; 4:00 gives 120°

Exercise 20 – Meeting halfway

\(\left( \frac{2 + 10}{2}, \frac{1 + 7}{2} \right) = (6, 4)\)

Total distance: \(\sqrt{8^{2} + 6^{2}} = \sqrt{100} = 10\) miles.

Each person travels half of 10 miles.

Meeting point (6, 4); Liam travels 5 miles

Key facts behind this basics practice solutions

Key facts

Point, line and plane are undefined terms; segments, rays and angles are defined from them.
Segment addition: if B is between A and C, then AB + BC = AC.
A midpoint at coordinate (a + b)/2 splits a segment into two congruent segments.
Angle addition: m∠ABD + m∠DBC = m∠ABC when ray BD is inside the angle.
A circle is all points in a plane at one fixed distance (the radius) from the center.

How to use this basics practice solutions

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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