This geometry foundations worksheet answer key solves all 23 problems of the points, lines, planes, segments and angles worksheet step by step.
Each solution shows the postulate or definition used, the equation, the algebra and a check. Compare every step with your own work, not only the final answer, and redraw any figure that did not match.
Geometry Foundations Worksheet Answer Key: problems 1–5
Problem 1: Naming points and rays
- P, Q and R lie on one line: P, Q, R.
- S is off the line, so S with any two of the others works: P, Q, S.
- Endpoint first: ray QR.
Problem 2: Defining a segment
A line segment is the part of a line that consists of two endpoints and all the points of the line between them.
Problem 3: Segment addition: find the whole
Segment Addition Postulate: \(AC=AB+BC=7+12=19\). AC = 19 in.
Problem 4: Segment addition: find a part
Subtract: \(BC=AC-AB=31-18=13\). BC = 13 cm. Check: \(18+13=31\).
Problem 5: Segment addition with algebra
\((3x+2)+(2x+7)=34\), so \(5x+9=34\) and \(5x=25\). x = 5.
\(AB=3(5)+2=17\) and \(BC=2(5)+7=17\). AB = 17 and BC = 17. Check: \(17+17=34\). B is the midpoint of \(\overline{AC}\).
Geometry Foundations Worksheet Answer Key: problems 6–10
Problem 6: Midpoint with algebra
A midpoint makes the halves equal: \(4x-3=2x+9\). Then \(2x=12\) and x = 6.
\(PM=4(6)-3=21\) and \(MQ=2(6)+9=21\). \(PQ=21+21=42\). PQ = 42.
Problem 7: Midpoint on a number line
\(AB=|10-(-4)|=14\). The midpoint is halfway: \(\frac{-4+10}{2}=\frac{6}{2}=3\). AB = 14; midpoint at 3. Check: \(3-(-4)=7\) and \(10-3=7\).
Problem 8: Angle addition: find the whole
\(m\angle ABC\)\(=\)\(m\angle ABD\)\(+\)\(m\angle DBC\)\(=38^{\circ}+47^{\circ}=85^{\circ}\). 85°.
Problem 9: Angle addition: find a part
\(m\angle BOC=112^{\circ}-65^{\circ}=47^{\circ}\). 47°. Check: \(65+47=112\).
Problem 10: Angle addition with algebra
\((4x+8)+(2x+4)=96\), so \(6x+12=96\) and \(6x=84\). x = 14.
\(m\angle PQS\)\(=4(14)+8=64^{\circ}\) and \(m\angle SQR\)\(=2(14)+4=32^{\circ}\). Check: \(64+32=96\). 64° and 32°.
Geometry Foundations Worksheet Answer Key: problems 11–15
Problem 11: Angle bisector
A bisector makes two equal angles: \(3x+5=5x-9\). Then \(14=2x\) and x = 7.
\(m\angle ABD\)\(=3(7)+5=26^{\circ}\) and \(m\angle DBC\)\(=5(7)-9=26^{\circ}\). \(m\angle ABC\)\(=26+26=52^{\circ}\). 52°.
Problem 12: Complement of an angle
\(90-63=27\). 27°.
Problem 13: Supplement of an angle
\(180-127=53\). 53°.
Problem 14: Complementary angles with algebra
\((2x+3)+(3x+7)=90\), so \(5x+10=90\) and \(5x=80\). x = 16.
The angles are \(2(16)+3=35^{\circ}\) and \(3(16)+7=55^{\circ}\). Check: \(35+55=90\). 35° and 55°.
Problem 15: Supplementary angles with algebra
\((4x-10)+(6x+20)=180\), so \(10x+10=180\) and \(10x=170\). x = 17.
The angles are \(4(17)-10=58^{\circ}\) and \(6(17)+20=122^{\circ}\). Check: \(58+122=180\). 58° and 122°.
Geometry Foundations Worksheet Answer Key: problems 16–20
Problem 16: Angles where two streets cross
Angles a and \(64^{\circ}\) are neighbors on a straight line, so \(a=180-64=116^{\circ}\).
Angle b is vertical to the \(64^{\circ}\) angle, so \(b=64^{\circ}\). Angle c is vertical to a, so \(c=116^{\circ}\). a = 116°, b = 64°, c = 116°.
Problem 17: Vertical angles with algebra
Vertical angles are congruent: \(7x-12=4x+18\). Then \(3x=30\) and x = 10.
\(m\angle 1\)\(=7(10)-12=58^{\circ}\) and \(m\angle 3\)\(=4(10)+18=58^{\circ}\). Angle 2 forms a linear pair with angle 1: \(m\angle 2\)\(=180-58=122^{\circ}\). 58° and 122°.
Problem 18: Perpendicular lines
Perpendicular lines form right angles, so \(3x+6=90\). Then \(3x=84\) and x = 28. Check: \(3(28)+6=90\).
Problem 19: Points and a circle
A circle is the set of all points in the plane at the same distance from the center. Only D is 6 cm from C. E is closer than 6 cm (inside) and F is farther (outside). Only point D is on the circle.
Problem 20: Road trip on a straight highway
Distance: \(74-18=56\) miles. Half of that is \(56: 2=28\) miles.
Rest stop: \(18+28=46\). 56 miles apart; mile marker 46. Check: \(74-46=28\).
Geometry Foundations Worksheet Answer Key: problems 21–23
Problem 21: Cable beside a utility pole
The pole and the ground are perpendicular, so the cable splits a right angle into two complementary angles.
\(90-73=28\). 17°.
Problem 22: Laptop lid
The new angle plus the \(40^{\circ}\) rotation makes the original angle (Angle Addition Postulate): \(115-40=75\). 75°.
Problem 23: Four angles at an intersection
A linear pair is supplementary: \((2x+15)+(3x-5)=180\), so \(5x+10=180\) and \(5x=170\). x = 34.
\(m\angle 1\)\(=2(34)+15=83^{\circ}\) and \(m\angle 2\)\(=3(34)-5=97^{\circ}\). Angle 3 is vertical to angle 1 and angle 4 is vertical to angle 2.
83°, 97°, 83°, 97°. Check: \(83+97=180\).
How to use this geometry foundations worksheet answer key
Read this geometry foundations worksheet answer key after trying each problem yourself, and redo any problem you missed.
See every chapter on the Geometry math page.