Practice the problems of Math in Focus Grade 7 Workbook Answer Key Chapter 7 Lesson 7.1 Constructing Angle Bisectors to score better marks in the exam.

## Math in Focus Grade 7 Course 2 B Chapter 7 Lesson 7.1 Answer Key Constructing Angle Bisectors

### Math in Focus Grade 7 Chapter 7 Lesson 7.1 Guided Practice Answer Key

**Hands-On Activity**

**Materials**

- ruler

Explore The Distance Between Points On The Sides Of An Ancle And Points On The Angle Bisector

Work in pairs.

\(\overrightarrow{Q S}\) is the angle bisector of ∠YQX. Points X and Y are equidistant from point Q.

Points S_{1}, S_{2}, S_{3}, and S_{4} lie on \(\overrightarrow{Q S}\), the angle bisector of ∠YQX.

Step 1.

Measure and record each length to the nearest tenth of a centimeter.

Answer:

S_{1}X = 5 cm S_{1}Y = 5 cm

S_{2}X = 4 cm S_{2}Y = 4 cm

S_{3}x = 3 cm S_{3}Y = 3 cm

S_{4}x = 2 cm S_{4}Y = 2 cm

Step 2.

Compare the lengths of \(\overline{S_{1} X}\) and \(\overline{S_{1} Y}\). Then compare the lengths of the two segments in each of the following pairs: \(\overline{S_{2} X}\) and \(\overline{S_{2} Y}\), \(\overline{S_{3} X}\) and \(\overline{S_{3} Y}\), and \(\overline{S_{4} X}\) and \(\overline{S_{4} Y}\). What do you observe about each pair of segment lengths?

**Math Journal**

Suppose you choose any point on the angle bisector. Do you think you will observe the same relationship between the lengths of the segments that connect the point to points X and V? What conclusion can you make?.

From the activity, any two points on the sides of an angle that are the same distance from the vertex are also the same distance from any point on the angle bisector.

Trace or copy ∠PQR. Then draw the angle bisector of ∠PQR.

Question 1.

Answer:

**Complete.**

Question 2.

Two walls intersect to form a right angle. In gym class, the students use the two walls to play a game in which the players line up so that each player is equidistant from the two walls.

a) Copy the diagram and draw a line to show where students should line up.

Answer:

We are given the angle:

The players are equidistant from the two walls if they are placed on the angles bisector. We buiLd the bisector.

First, with the compass point at the vertex, draw an arc that intersects \(\overrightarrow{Q P}\) and \(\overrightarrow{Q R}\).

Label the intersection points as A and B.

Using the same radius, draw an arc with A as the center.

Using B as the center, draw another arc with the same radius. Label the point where the two arcs intersect as C.

Use a straightedge to draw \(\overline{Q C}\).

b) What is the measure of the angle formed by the students and Wall 1 ? How do you know?

Answer:

Because the measure of the angle PQR is 90° and the bisector divides it in two congruent parts, the measure of the angle formed by the students and each of the walls is:

\(\frac{90^{\circ}}{2}\) = 45°

**Trace or copy the diagram. Then complete.**

Question 3.

Use ∠X to construct a 15° angle whose vertex is point X.

Answer:

### Math in Focus Course 2B Practice 7.1 Answer Key

**Construct the angle bisector of ∠ABC on a copy of each figure.**

Question 1.

Answer:

Question 2.

Answer:

**Draw each angle with a protractor. Then construct its angle bisector.**

Question 3.

m∠POR = 75°

Answer:

Question 4.

m∠ADE = 122°

Answer:

**Copy the angle shown. Then perform the indicated construction.**

Question 5.

Construct a 25° angle at point X.

Answer:

Question 6.

Construct a 108° angle at point Y.

Answer:

**Solve.**

Question 7.

Justin wants to construct the angle bisector of ∠XYZ. Trace or copy the diagram. Using only a compass and a straightedge, construct the angle bisector and describe each step clearly.

Answer:

Explanation:

1. With ‘Y’ as center draw an arc of any radius to cut the rays of the angle at X and Z.

2. With ‘Z’ as center draw an arc of radius more than half of XZ, in the interior of the given angle.

3. With ‘X’ as center draw an arc of same radius to cut the previous arc at ‘o’.

4. Join YO. YO is the angle bisector of the given angle.

Question 8.

Draw two straight lines intersecting at an angle of 108°. Find the points that are equidistant from the two sides of each 108° angle formed by the intersecting lines.

Answer:

We are given:

The points equidistant from the two sides of each 108° angle formed by the intersection of the two Unes are placed on the 108° angles bisector.

We construct the bisector of the angle NOQ.

With the compass point at the vertex O, draw an arc that intersects \(\overrightarrow{O N}\) and \(\overrightarrow{O Q}\).

Label the intersection points as A and B.

Using the same radius, draw an arc with B as the center.

Using A as the center, draw another arc with the same radius Label the point where the two arcs intersect as C.

Use a straightedge to draw \(\overline{O C}\), which is the bisector of both vertical angles NOQ and POM.

Question 9.

Draw an obtuse angle of any measure and label it as ∠XYZ. Construct an angle that is one-fourth the measure of ∠XYZ, describing briefly the steps involved.

Answer:

We are given the obtuse angle:

In order to divide the angle in 4 congruent parts, first we construct the bisector \(\overrightarrow{Y V}\) of

the angle XYZ:

We have:

m∠XYV = m∠VYZ = \(\frac{m \angle X Y Z}{2}\)

Then we construct the bisector \(\overrightarrow{Y W}\) of the angle XYV:

We have:

m∠XYW = m∠WYV = \(\frac{m \angle X Y V}{2}\)

= \(\frac{\frac{X Y X}{2}}{2}\) = \(\frac{m \angle X Y Z}{4}\)

Divide the angle twice, constructing angle bisectors.

Question 10.

The diagram shows ∠ABC with \(\overrightarrow{B D}\) being its angle bisector and BF = BG E is a point on \(\overrightarrow{B D}\) and m∠ABD = 26°. Copy and complete.

m∠DBC =

Length of \(\overline{E G}\) = Length of

Answer:

\(\overline{B D}\) is angle bisector:

m∠DBC = m∠ABD

Substitute:

m∠DBC = 26°

△BEF ≅ △BEG (side-angle-side)

BF = BG

m∠DBC = m∠ABD

BE = BE

Congruent triangles:

Length of \(\overline{E G}\) = Length of \(\overline{E F}\)

Question 11.

Officials are planning to build a new airport to serve three major cities in one region. The cities are located at W, X, and V, which are represented by the vertices of the triangle shown. The officials want to place the airport at the intersection of the angle bisectors of ∠XWY and ∠XYW. Copy or trace the triangle. Find the possible location of the airport and label it point Q.

Answer:

We construct the bisector of ∠XWY:

We construct the bisector of ∠XYW:

The intersection Q of the two bisectors is the possible location for the airport.

Question 12.

Joshua used a square piece of paper to make a paper airplane. As a first step, he made several folds in the paper, as shown in the diagram. First he folded along diagonal \(\overline{\mathrm{QS}}\), then he unfolded the paper. Next he folded along \(\overline{\mathrm{QT}}\) so that \(\overline{\mathrm{PQ}}\) lined up with \(\overline{\mathrm{QS}}\). Then he unfolded the paper again. In the diagram, what is the measure of ∠PQT? Give a brief explanation for your answer.

Answer:

In order to Line up \(\overline{P Q}\) with \(\overline{Q S}\) folding along \(\overline{Q T}\), the points of \(\overline{Q T}\) should be equidistant to \(\overline{P Q}\) and \(\overline{Q S}\), therefore on the bisector of ∠PQS.

we aetermine m∠PQT:

m∠PQT = \(\frac{m \angle P Q S}{2}\)

= \(\frac{\frac{90^{\circ}}{2}}{2}\) = \(\frac{45^{\circ}}{2}\)

= 22.5°

Question 13.

Kimberly wants to bisect the straight angle shown, and then bisect one of the resulting angles. She then wants to continue this process until she obtains an angle of measure 11.25°. How many times does she need to construct an angle bisector to produce an angle with this measure? Explain.

Answer:

Given the Angle measure of 11.25°.

Here we need to find out the angle which after bisecting gives an angle of 11.25°.

We know that when an angle is bisected it is divided into two equal angles.

Therefore, the double of the angle obtained after bisecting we will get the angle that was bisected

Hence, the angle is

= 2 × 11.25°

= 22.5°

Question 14.

Max designed a support for a bridge. In his design, five spokes are attached to a metal beam. The angles formed by the spokes all have the same measure. Explain how Max can use geometric construction to accurately draw his design.

Answer:

Take an arbitrary point outside the line:

Draw an arc with the point in A. Label the intersections with the Line by B and C:

Join A to B and C:

Construct the bisector of ∠BAC:

Construct the bisectors of ∠BAD and ∠DAC: