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1. Foundations of Geometry
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Geometry

1.05 Angles and constructions

Lesson
Practice
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Worksheet
What do you remember?
1

Name the angles shown in three different ways.

a
Segments P Q and Q R. The angle formed by P Q and Q R is marked.
b
Segments X Y and Y Z. The angle formed by X Y and Y Z is marked.
2

For each angle shown

i

Identify the measure of the angle

ii

Use the measure to classify the angle as acute, obtuse, right, or straight

a
An angle drawn on a diagram of a protractor. The vertex of the angle is at the center point of the protractor. A ray is aligned with the 0 mark, and the other ray is aligned with the 50 mark. An arrow is pointing from the ray at the 0 mark to the ray at the 50 mark.
b
An angle drawn on a diagram of a protractor. The vertex of the angle is at the center point of the protractor. A ray is aligned with the 0 mark, and the other ray is aligned with the 160 mark. An arrow is pointing from the ray at the 0 mark to the ray at the 160 mark.
3

Find the measure of each angle

a

If m \angle IJK = 150 \degree, find m \angle BJK

Adjacent angles I J B and B J K. Angle I J B has a measure of 25 degrees.
b

Find m \angle KLM

Adjacent angles B L K and B L M. Angle B L K has a measure of 30 degrees, and angle B L M has a measure of 60 degrees.
4

Use the diagrams to identify two pairs of congruent angles.

a
Lines A D and F C intersecting at point G. Point A is in the interior of angle B G F, and point D is in the interior of angle C G E. A line is drawn from A to D passing through G. Angle F G A has a measure of 50 degrees, angle A G B has a measure of 40 degrees, angle C G D has a measure of 50 degrees, and angle D G E has a measure of 40 degrees.
b
Lines H F and J G intersecting at point Z. Ray Z A is in the interior of angle J Z F. Angles J Z H and F Z G have the same angle decorations, as well as angles J Z F and H Z G.
5

Use the diagram to find each angle measure

Four rays sharing a common endpoint E drawn on a diagram of a protractor. Point E is at the center point of the protractor. Ray A E is aligned with the 0 mark, ray B E is aligned with the 50 mark, ray C E is aligned with the 105 mark, and ray D E is aligned with the 160 mark.
a
m \angle AEB
b
m \angle AED
c
m \angle CED
d
m \angle DEB
6

Determine whether each of the following diagrams shows the given angle pair. Explain.

a

Complementary angles

Two adjacent angles measuring 12 degrees, and 68 degrees.
b

Vertical angles

Two intersecting lines, forming a pair of opposite angles, both measuring 28 degrees.
c

Linear pair

A line and a segment with one of the endpoints of the segment on the line, and the other endpoint not on the line. Two adjacent angles are formed, measuring 109 degrees, and 71 degrees.
d

Linear pair

A line and a segment with one of the endpoints of the segment on the line, and the other endpoint not on the line. Two adjacent angles are formed, measuring 137 degrees, and 43 degrees.
7

Identify the missing angle for the angle pair.

a

⬚ forms a linear pair with \angle{QOT}.

Line M U, segment P O, and segment O S intersecting at point O. Point N is between M and O, point T between O and U, point Q between P and O, and point R between O and S. Angle Q O T is a right angle.
b

⬚ forms a vertical pair with \angle{BEH}.

Two lines A H and B I intersecting at point E. Point D is between E and B, point G between E and I, point C between A and E, and point F between E and H. Angle B E H is marked.
8

Each statement shown has an error. Identify the error and rewrite the statement correctly.

a
\angle ABC = 90 \degree
Right triangle A B C with right angle A B C.
b

In \angle XZY, \, Y is the vertex.

c

180 \degree is an obtuse angle since its measure is greater than 90 \degree.

d

If point P lies in the interior of \angle JKL then m \angle JKP + m\angle JKL = m\angle LKP.

9

Determine the value of the variable in each of the following diagrams:

a
Adjacent angles H G K and K G J and a point F outside both angles. A segment is drawn from G to F. Angle H G J is a right angle. Angles H G K has a measure of 63 degrees, and K G J has a measure of a degrees.
b
Line E C with point D on the line, and points A and B not on the line. Segments are drawn from D to A, and D to B. Segment A D is in the interior of angle B D E. Angle A D E has a measure of 80 degrees, A D B has a measure of m degrees, and B D C has a measure of 49 degrees.
Let's practice
10

Use the diagram to find an example of each angle pair if it exists:

a

Linear pair

b

Vertical angles

c

Complementary angles

d

Supplementary angles that are not also a linear pair

Line Y U with point W on the line, points V, T, X, and Z are not on the line. Points V and T on one side of the line, and points X and Z on the other side of the line. Segments from W to V, from W to T, from W to Z, and from W to X are drawn. Segment V W is in the interior of angle Y W T, and X W in the interior of angle Z W U. Angles T W U and Z W X are right angles.
11

Use the angle addition postulate to find the indicated angle

a

m \angle PQR = 145 \degree

Find m\angle{SQR}

Adjacent angles S Q P and S Q R. Angle S Q P has a measure of 3 x plus 7 degrees, and S Q R has a measure of 2 x minus 2 degrees.
b

m \angle ADC = 120 \degree

Find m\angle{ADB}

Adjacent angles A D B and B D C. Angles A D B and B D C are congruent. Angle B D C has a measure of 4 x minus 23 degrees.
c

m \angle ADC = (10x+35) \degree

Find m\angle{ADB}

Adjacent angles A D B and B D C. Angle B D C is a right angle. Angle A D B has a measure of 5 x minus 15 degrees.
d

m \angle ADC = 70 \degree

Find m\angle{ADB}

Adjacent angles A D B and B D C. Angle B D C has a measure of x minus 9 degrees, and angle A D B has a measure of 5 x plus 1 degrees.
12

For each diagram:

i

Write an equation that models the relationship shown in the diagram.

ii

State the theorem or postulate that justifies this equation.

iii

Solve for x.

a
Two intersecting lines with opposite angles measuring 10 x plus 1 degrees, and 12 x minus 5 degrees
b
Two intersecting lines with adjacent angles measuring 4 x plus 7 degrees, and 2 x plus 5 degrees.
13

Use the given information to solve for the value of x.

a

\overrightarrow{RQ} bisects \angle{PRS}

Rays P R, R Q, and R S forming two adjacent angles, P R Q and Q R S. Angle P R Q has a measure of x plus 40 degrees, and Q R S has a measure of 3 x minus 20 degrees.
b

\overrightarrow{DB} bisects \angle{ADC} and \\m\angle{ADC}=138\degree

Rays A D, D B, and D C forming two adjacent angles, A D B and B D C. Angle B D C has a measure of 4 x minus 23 degrees.
14

The measure of an angle's complement is five less than half the measure of the angle's supplement. Find the measure of the angle.

15

Use the given information to find the measures of the indicated angles:

  • m\angle{1}=2x-30
  • m\angle{2}=5x-120
  • m\angle{3}=10x-150

  • \angle{1} and \angle{3} form a linear pair

  • \angle{1} and \angle{2} are vertical angles

a
m\angle{1}
b
m\angle{3}
16

Given that m \angle ABC=120 \degree , \, \angle ABC is bisected by \overrightarrow{BD}, and \angle ABD is bisected by \overrightarrow{BE}. Draw a diagram and find m \angle EBC.

17

Construct \angle{X} so that \angle{B} \cong \angle{X}.

Segments A B and B C, and a point X not on both segments. The angle formed by A B and B C is an obtuse angle.
18

Construct \overrightarrow{BD} so that \overrightarrow{BD} bisects \angle{B}.

Segments A B and B C.
19

Use proper geometric notation to clearly describe each construction step shown.

An acute angle P N O.

Step 1:

An acute angle P N O with an arc drawn with the center at P. The arc intersects the segment N P on point R, and intersects the segment N O on point S.

Step 3:

An acute angle P N O with point R on segment N P, and point S on segment N O. Two arcs with the same radius are drawn. One has a center at R, and one has a center at S. The two arcs intersect at a point T in the interior of angle P N O.

Step 2:

An acute angle P N O with point R on segment N P, and point S on segment N O. An arc with center at S is drawn.

Step 4:

An acute angle P N O with point R on segment N P, point S on segment N O, and point T in the interior of angle P N O. A line connects N and T.
Let's extend our thinking
20

Two students are debating over the answer to the following flawed question:

Find m\angle{Q}

Segments P Q, S Q, and R Q intersecting at point Q. Q S is in the interior of the angle formed by P Q and Q R. The angle formed by P Q and S Q is labeled 37 degrees, and the angle formed by S Q are R Q is labeled 22 degrees.

Presley:

The m\angle{Q} should be 59\degree

Qing:

The m\angle{Q} should be 22\degree

Determine the issue with this question and explain why the two students chose different answers.

21

Determine if the statement is always, sometimes, or never true. Create diagrams to justify your answers.

a

Complementary angles are also adjacent.

b

A linear pair is supplementary.

c

An obtuse angle forms a linear pair with an acute angle.

d

Vertical angles form a linear pair.

e

Vertical angles are supplementary.

22

Determine if the information given is enough to justify the conclusion. Draw a diagram to support your solution.

Given: \angle MNP and \angle PNR form a linear pair and \overline{NP} bisects \angle{MNR}.

Conclusion: m \angle MNP = m \angle PNR = 90 \degree

23

Construct an angle twice the size of \angle XYZ with a vertex at point P. Describe each step of your construction.

An acute angle X Y Z, and a point P outside the angle.
24

Construct an angle whose measure is equal to m\angle ABC + m\angle DEF.

An acute angle A B C, and an obtuse angle D E F.
25

Describe a sequence of constructions that would create a 45 \degree angle.

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