Cycle1.Lesson3: Construction of Congruent Angles & Segments

Math Response

Find the length of $$\overline{AC}$$

Type answer here

Correct Answer: 64

Deg^°

Rad

Match

Match the following pictures with the terms below.

• 1

• 2

• 3

• 4

• 5

Acute angle

Reflex angle

Obtuse angle

Right Angle (90 degrees)

Straight angle

• Acute angle

• Reflex angle

• Obtuse angle

• Right Angle (90 degrees)

• Straight angle

LO: SWBAT construct congruent segments,
congruent angles, a segment bisector, and an
angle bisector using a compass and a straightedge.

DOL: Given 5 problems, students will correctly
construct congruent segments, congruent angles, a
segment bisector, and an angle bisector using a
compass and a straightedge in at least 4 of 5
questions.

GEOM.5B Construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular
bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge.

3

4

Lesson Vocabulary

Congruent Segments

Two segments are congruent
(≅) if they have the same
length.

Segment Bisector

A segment bisector is a line, line
segment, ray, or point that cuts a
line segment exactly in half.

Angle Bisector

A ray, line, or segment that divides
an angle into two equal parts.

Congruent Angles

Two angles are congruent (≅) if
they have the same measures.

Angle Addition Postulate

The sum of two adjacent angle
measures will equal the angle
measure of the larger angle that
they form together.

Adjacent Angles

Angles that share a common
vertex and common side.

5

Lesson Vocabulary

Linear Pair

A linear pair are two
adjacent angles and the
measures add up to
180°.

Linear Pair Postulate

If two angles form a linear pair
then the measures of the angles
add up to 180°

Vertical Angles

Anglesformed by two
intersecting lines or line
segments. They share a
common vertex and are
congruent.

6

Revisiting Types of Angles

Acute Angle

Reflex Angle

Obtuse Angle

Right Angle

Straight Angle

An angle with a
measure that is
equal to 180
degrees.

An angle with a
measure that is
equal to 90
degrees.

An angle with a
measure that is
greater than 90
but less than 180
degrees.

An angle with a
measure that is
greater than 180
but less than 360
degrees.

An angle with a
measure that is
greater than 0
but less than 90
degrees.

Define or describe each angle.

7

Naming Angles

Names

∠Y
∠XYZ
∠ZYX

Names

∠B
∠ABC
∠CBA

Names

∠Q
∠PQR
∠RQP

Name the angle

Name the angle

Two ways to name an angle.

1.Use one capital letter (vertex)

2Use three capital letters with the vertex in the middle

​3 min

8

Steps in Constructing Congruent Angles

​7 min

​https://www.mathspad.co.uk/i2/construct.php

Reorder

Reorder the following steps for constructing congruent angles.

1

2

3

4

• 1
• 2
• 3
• 4

10

Midpoint: Segment Bisector

In the previous lesson, we learned about Midpoint as the point that divides a segment into
two congruent segments as shown in Figure A. It is also known as a segment bisector.

A segment bisectoris a line, line
segment, ray, or point that cuts a
line segment exactly in half.

Vertical Angles are formed by two
intersecting lines/segments. They share a
common vertex and are congruent.

●Identify the segment
bisectors in each figure.

●Which figures have pairs of vertical
angles?

Figure D

​3 min

Poll

Vertical Angles are congruent $$\left(\cong\right)$$ .

• True

• False

11

Steps in Constructing Angle Bisector

1. Use a straightedge to draw the angle and label it.

Summary of the construction steps.

2. Place the pointer of the compass on the vertex. Swing it to draw an arc

that intersects both sides.

3. Move the pointer to a side which the arc intersects and make a second arc

mark on the interior of the angle. Repeat the process on the other side.
Make sure the arcs intersects.

4. Connect the vertex to the intersection of the arcs.

​7 min

Reorder

Reorder the following steps of constructing an angle bisector.

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

Categorize

Options (6)

Organize these pictures based off the angle created by their bisector.

Obtuse Angle

Acute Angle

Right Angle

Straight Angle

14

Angle Pairs Formed by Angle Bisectors

Adjacent Angles are angles
that share a common vertex
and common side.

●Are there examples of
adjacent angles below?

Linear Pair are adjacent angles
that add up to 180 degrees.

●Are there examples of linear
pair below?

Adjacent Angles

Linear Pair

​3 min

15

Angle – Addition Postulate

F

F

F

F

F

Angle-Addition Postulate states that the sum of the measure of
two adjacent angles is equal to the measure of the angles they
form.

​2 min

17

Application Questions

Use algebra to determine the value of the variable and determine the
measure of the angles in each figure.

6.

5.

​5 min

18

Steps in Constructing Congruent Segments

1. Use a straightedge to draw a reference ray and label the endpoint.

Summary of the construction steps.

2. Open or set the compass to the length of the segment you want to copy.

3. With the same compass setting, put the compass point on point the endpoint of your

reference ray, and draw an arc that intersects the ray, label the point of intersection.

​5 min

Reorder

Reorder the following steps to construct congruent line segments.

1

2

3

• 1
• 2
• 3

23

Application Question

Apply what you learned about congruent angles and congruent segments to answer the
following questions about the diagram on the right. Justify your answer if possible.

1.

Name the angle bisector of ∠TPR.

2.

Name two pairs of congruent segments.

3.

What is the m∠QPR?.

4.

What is the m∠RPS?

5.

What is the m∠QPV?

​5 min

24

Application Questions

Decide whether each statement is sometimes, always, or never true. Justify your answer
if possible.

1.

Every angle has exactly one angle bisector.

2.

Congruent angles have the same measure.

3.

Congruent segments have the same measure.

4.

If two angles are adjacent then they are a linear pair.

5.

Vertical angles are adjacent angles.

​5 min

Demonstration of Learning

DOLGiven 5
problems, students
will correctly
construct
congruent
segments,
congruent angles,
a segment
bisector, and an
angle bisector
using a compass
and a straightedge
in at least 4 of 5
questions.

25