# Geometry Worksheets - Page 2

Geometry Worksheets
• Page 2
11.
Name the property that justifies the statement.
$x$ = $x$, where $x$ is a real number.
 a. substitution property b. transitive property c. symmetric property d. reflexive property

#### Solution:

Every number is equal to itself.
[Reflexive Property.]

x = x shows the Reflexive Property of equality.

12.
Which statement illustrates the symmetric property?
 a. $p$ × $\frac{1}{p}$ = 1 b. $p$ + $q$ = $r$ c. If $p$ + $q$ = $r$, then $r$ = $p$ + $q$. d. If $p$ + $q$ = $r$ and $r$ = $m$ + $n$, then $p$ + $q$ = $m$ + $n$.

#### Solution:

The symmetric property states that if a = b, then b = a.

So, if p + q = r, then r = p + q.

13.
If m$\angle$A + m$\angle$B = 90, then what all can be concluded from this?
(i) One arm of $\angle$A and one arm of $\angle$B are perpendicular
(ii). $\angle$A and $\angle$B are complementary.
(iii). m$\angle$A = 90 - m$\angle$B
(iv) m$\angle$B = 90 - m$\angle$A
(v). $\angle$A and $\angle$B cannot be the angles of the same triangle.
 a. (ii), (iii), (iv) and (v) b. All statements c. (ii), (iii) and (iv) d. (i), (ii) and (v)

#### Solution:

The two angles can be with two different arms. Statement (i) need not be correct.

The two angles can be the angles of the same triangle since sum of the angles of a triangle is 180 degrees.

(ii), (iii) and (iv) are correct.

14.
Select the correct statements.
(i). If two angles are supplements of congruent angles, then the two angles are supplementary.
(ii). If two angles are complements of congruent angles, then the two angles are complementary.
(iii). If two congruent angles are complement to each other, then each is a right angle.
(iv). If two congruent angles are supplement to each other, then each is a right angle.
 a. (i) and (ii) only b. (ii) and (iv) only c. (i) and (iii) only d. (iv) only

#### Solution:

If two congruent angles are supplement to each other, then each has to be a right angle.
[90 + 90 = 180.]

Only statement (iv) is correct.

15.
Sum of the measures of the angles in a triangle is equal to 180. Select the conclusions that can be made out of this.
(i). Sum of the measures of the angles in a quadrilateral = 360
(ii). A triangle can have only one angle which is equal to or more than 90 degrees.
(iii). Two angles of a triangle can never be complementary.
(iv). Two angles of a triangle can never be supplementary
 a. (i), (ii), and (iv) b. (ii) and (iv) c. (ii), (iii), and (iv) d. (i), (ii), and (iii)

#### Solution:

A quadrilateral can be divided into two triangles by a diagonal. Sum of the angles in a quadrilateral = sum of the angles of two triangles = 360.

There cannot be two angles equal to or more than 90 degrees in a triangles as the sum of the measures of all the three angles is 180.

Complementary means sum = 90. Two angles in a triangle can be complementary.

Two angles of a triangle cannot be supplementary, because sum of the measures of all the three angles is 180.

Statements (i), (ii), and (iv) are correct.

16.
In the diagram, if $a$ = 114o, identify a relation between $x$ and $a$.

 a. $a$ > $x$ b. $a$ < $x$ c. $a$ = $x$ d. cannot be determined

#### Solution:

xo + 114o = 180o

xo = 180o - 114o
[Subtraction Property.]

xo = 66o
[Subtract.]

So, 114o > xo
[Since 114o > 66o.]

17.
Use the figures to decide which type of angle pair does $x$o and $y$o represent.

 a. Straight angles b. Vertical angles c. Complementary angles d. Supplementary angles

#### Solution:

xo + 60o = 180o
[From the figure.]

So, xo = 120o
[Use subtraction property.]

yo + 30o = 90o
[From the figure.]

So, yo = 60o
[Use subtraction property.]

xo + yo = 120o + 60o = 180o

So, the two angles xo and yo are supplementary angles.

18.
In the diagram, if $a$ = 25o, then choose a relation between $x$ and $a$.

 a. cannot be determined b. $x$ < $a$ c. $x$ > $a$ d. $x$ = $a$

#### Solution:

mAOB = mAOC + mCOB = x + 25

mAOB = 90
[From the diagram.]

So, x + 25o = 90o

x = 90o - 25o
[Subtraction Property.]

x = 65o
[Subtract.]

So, x > 25
[65o > 25o.]

19.
Solve for $x$ and $y$.

 a. $x$ = 39, $y$ = 34 b. $x$ = 34, $y$ = 39 c. $x$ = 26, $y$ = 51 d. $x$ = 26, $y$ = 102

#### Solution:

3x =78
[Vertical angles.]

x = 26
[Divide by 3, Division property.]

2y + 78 = 180
[Supplementary angles.]

2y + 78 - 78 = 180 - 78
[Subtraction property.]

2y = 102
[Simplify.]

y = 51
[Division property.]

x = 26, y = 51

20.
Solve for $x$, $y$, and $z$.

 a. $x$ = 18, $y$ = 9, $z$ = 23 b. $x$ = 4, $y$ = 6, $z$ = 2 c. $x$ = 16, $y$ = 7, $z$ = 47 d. $x$ = 1, $y$ = 11, $z$ = 38

#### Solution:

8y - 17 = 5y + 4
[Vertical angles.]

3y = 21, y = 7
[Simplify.]

8x + 13 + 5y + 4 = 180
[Supplementary angles.]

8x + 13 + 35 + 4 = 180
[Substitution property.]

8x = 128, x = 16
[Subtraction and division properties.]

3z = 8x + 13
[Vertical angles.]

3z = 128 + 13 = 141
[Substitution property.]

z = 47
[Simplify using division property.]