Geometry Proof Basics Reference Sheet
Geometry Reference
Points, Segments & Rays
Page 1 of 4
1 MIDPOINT
Segment Division
G M K equal equal M is in the center
GIVEN
\(M\) is the midpoint of \(\overline{GK}\)
Plain Definition
A midpoint is a point on a line segment that separates it into two congruent segments of equal length.
CONCLUSION
\(\overline{GM} \cong \overline{MK}\) Lengths are equal: \(GM = MK\)
REASON
Definition of Midpoint
2 SEGMENT BISECTOR
Segment Division
T P A B Z
GIVEN
\(\overleftrightarrow{AB}\) bisects \(\overline{TP}\)
Plain Definition
A segment bisector is a line, ray, or segment that passes through the midpoint, separating it into two congruent segments.
CONCLUSION
\(\overline{TZ} \cong \overline{ZP}\) Point \(Z\) is the midpoint of \(\overline{TP}\)
REASON
Definition of Segment Bisector
3 ANGLE BISECTOR
Angle Division
W J F C 7 3
GIVEN
\(\overrightarrow{WC}\) bisects \(\angle FWJ\)
Plain Definition
An angle bisector is a ray that divides an angle into two congruent adjacent angles of equal measure.
CONCLUSION
\(\angle 3 \cong \angle 7\) Measures are equal: \(m\angle 3 = m\angle 7\)
REASON
Definition of Angle Bisector
Geometry Reference
Complementary & Perpendicular
Page 2 of 4
4 COMPLEMENTARY ANGLES (DEF 1)
Sum = 90°
MEASURES ADD TO 90° ∠5 ∠12
GIVEN
\(\angle 5\) is complementary to \(\angle 12\)
Plain Definition
Complementary angles are two angles whose measures have a sum of exactly 90° (they do not need to touch).
CONCLUSION
\(m\angle 5 + m\angle 12 = 90^\circ\) Sum of the angle measures equals 90°
REASON
Definition of Complementary Angles
5 COMPLEMENTARY ANGLES (DEF 2)
Adjacent Right Angle
K J M N ∠1 ∠4
GIVEN
\(\angle 1\) is complementary to \(\angle 4\)
Plain Definition
Two adjacent angles that form a right angle together are complementary and sum to 90°.
CONCLUSION
\(\angle JKM\) is a right angle The combined angle measures exactly 90°
REASON
Definition of Complementary Angles
6 PERPENDICULAR LINES
Right Angle Intersection
X P K Y A
GIVEN
\(\overleftrightarrow{XP} \perp \overleftrightarrow{KY}\)
Plain Definition
Perpendicular lines are two intersecting lines, rays, or segments that meet to form right angles (90°).
CONCLUSION
\(\angle PAK\) is a right angle Measure is 90°: \(m\angle PAK = 90^\circ\)
REASON
Definition of Perpendicular Lines
Geometry Reference
Supplementary & Straight Angles
Page 3 of 4
7 SUPPLEMENTARY ANGLES (DEF 1)
Sum = 180°
MEASURES ADD TO 180° ∠11 ∠8
GIVEN
\(\angle 11\) is supplementary to \(\angle 8\)
Plain Definition
Supplementary angles are two angles whose measures have a sum of exactly 180° (they do not need to touch).
CONCLUSION
\(m\angle 8 + m\angle 11 = 180^\circ\) Sum of the angle measures equals 180°
REASON
Definition of Supplementary Angles
8 SUPPLEMENTARY ANGLES (DEF 2)
Straight Angle Pair
S U T V ∠SUV ∠TUV
GIVEN
\(\angle SUV\) is suppl. to \(\angle TUV\)
Plain Definition
Two adjacent angles that form a straight angle (line) together are supplementary and sum to 180°.
CONCLUSION
\(\angle SUT\) is a straight angle The combined angle measures exactly 180°
REASON
Definition of Supplementary Angles
9 STRAIGHT ANGLE
Angle Measurement
S 180° Opposite Rays form a Line
GIVEN
\(\angle S\) is a straight angle
Plain Definition
A straight angle is an angle whose sides form opposite rays in a flat line and whose measure is exactly 180°.
CONCLUSION
\(m\angle S = 180^\circ\) Measure is equal to exactly 180°
REASON
Definition of Straight Angle
Geometry Reference
Angle Classifications
Page 4 of 4
10 RIGHT ANGLE
Exactly 90°
R 90° Square Corner
GIVEN
\(\angle R\) is a right angle
Plain Definition
A right angle is an angle whose measure is exactly 90°. Its perpendicular rays form a square corner.
CONCLUSION
\(m\angle R = 90^\circ\) Measure is equal to exactly 90°
REASON
Definition of Right Angle
11 ACUTE ANGLE
Between 0° and 90°
C 90° ref < 90° Smaller than 90°
GIVEN
\(\angle C\) is an acute angle
Plain Definition
An acute angle is an angle whose measure is strictly less than 90° (and greater than 0°).
CONCLUSION
\(0^\circ < m\angle C < 90^\circ\) Measure is between 0° and 90°
REASON
Definition of Acute Angle
12 OBTUSE ANGLE
Between 90° and 180°
B 90° ref > 90° < 180°
GIVEN
\(\angle B\) is an obtuse angle
Plain Definition
An obtuse angle is an angle whose measure is greater than 90° and less than 180°.
CONCLUSION
\(90^\circ < m\angle B < 180^\circ\) Measure is strictly between 90° and 180°
REASON
Definition of Obtuse Angle
Geometry Proof Deductions Worksheet
Student Practice
Points, Segments & Rays
Page 1 of 4
Name:
Date:
Period:
1 MIDPOINT
Fill in Conclusion & Reason
G M K M is in the center
GIVEN
\(M\) is the midpoint of \(\overline{GK}\)
Definition Clue
A midpoint separates a segment into two congruent segments of equal length.
CONCLUSION (C) State congruent segments:
REASON (R) State geometric theorem/def:
2 SEGMENT BISECTOR
Fill in Conclusion & Reason
T P A B Z
GIVEN
\(\overleftrightarrow{AB}\) bisects \(\overline{TP}\)
Definition Clue
A line that crosses a segment at its midpoint, dividing it into two congruent parts.
CONCLUSION (C) State congruent segments:
REASON (R) State geometric theorem/def:
3 ANGLE BISECTOR
Fill in Conclusion & Reason
W J F C 7 3
GIVEN
\(\overrightarrow{WC}\) bisects \(\angle FWJ\)
Definition Clue
A ray that divides an angle into two congruent adjacent angles of equal measure.
CONCLUSION (C) State congruent angles:
REASON (R) State geometric theorem/def:
Student Practice
Complementary & Perpendicular
Name: Page 2 of 4
4 COMPLEMENTARY ANGLES (DEF 1)
Fill in Conclusion & Reason
MEASURES ADD TO 90° ∠5 ∠12
GIVEN
\(\angle 5\) is complementary to \(\angle 12\)
Definition Clue
Two angles whose measures have a sum of exactly 90° (they do not need to touch).
CONCLUSION (C) Write angle measure equation:
REASON (R) State geometric theorem/def:
5 COMPLEMENTARY ANGLES (DEF 2)
Fill in Conclusion & Reason
K J M N ∠1 ∠4
GIVEN
\(\angle 1\) is complementary to \(\angle 4\)
Definition Clue
Two adjacent angles that form a combined right angle together are complementary.
CONCLUSION (C) Identify combined angle:
REASON (R) State geometric theorem/def:
6 PERPENDICULAR LINES
Fill in Conclusion & Reason
X P K Y A
GIVEN
\(\overleftrightarrow{XP} \perp \overleftrightarrow{KY}\)
Definition Clue
Two lines, rays, or segments that intersect to create right angles (90°).
