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Chapter 6: Q11. (page 207)

Write proofs in two-column form.

Given: The diagram. Prove: m鈭�1>m鈭�4

Short Answer

Expert verified

Statement

Reasons

1. , m鈭�1>m鈭�2, m鈭�2>m鈭�3

Exterior angle theorem

2.m鈭�1>m鈭�3

Property of Inequality If and then .

3.m鈭�3=m鈭�4

Vertically opposite angle.

4.m鈭�1>m鈭�4

Substitution property

Step by step solution

01

Step 1. Draw diagram and define exterior angle inequality theorem

Two triangles as shown below are the given triangles.

The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

Therefore, from the figure m鈭�1>m鈭�2and m鈭�2>m鈭�3.

02

Step 2. Property of inequality and use vertically opposite angle property

If a>band b>cthen a>c. Since m鈭�1>m鈭�2and m鈭�2>m鈭�3, therefore m鈭�1>m鈭�3.

The vertically opposite angles are always equal.

Since 鈭�3and 鈭�4are vertically opposite angles, therefore m鈭�3=m鈭�4.

03

Step 3. Prove the statement

Sincem鈭�3=m鈭�4 andm鈭�1>m鈭�3 therefore by substitution property role="math" localid="1637858624737" m鈭�1>m鈭�4.

Hence proved.

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Most popular questions from this chapter

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write Yes or No).

Given: Point Y lies between points X and Z

a.XY=12XZ

b.XZ=XY+YZ

c.XZ>XY

d.YZ>XY

e.XZ>YZ

f.XZ>2XY

For each statement in Ex 5-10 copy and complete a table like the one shown below.

If x2>1, then x>1.

Classify the following conditional as true or false.

If a>b,b=c, then a>c.

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write Yes or No).

Given: Point Y lies between points X and Z

a. XY=12XZ

b. XZ=XY+YZ

c. XZ>XY

d. YZ>XY

e. XZ>YZ

f.XZ>2XY

Arrange sentences (a)-(e) in an order that completes an indirect proof of the following statement: In a plane, two lines perpendicular to a third line are parallel to each other.

Given: Lines a, b and t lie in a plane; a鈯�t;b鈯�t

Prove: a鈥�b

(a) Then a intersects b in some point Z.

(b) But this contradicts the theorem which says that there is exactly one line perpendicular to a given line through a point outside the line.

(c) It is false that a is not parallel to b, and it follows that a鈥�b

(d) Assume temporarily that a is not parallel to b.

(e) Then there are two lines through Z and perpendicular to t.
See all solutions

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