Exercise 128 Page 424

a Consider the given diagram.

It depicts a quadrilateral with a pair of opposite sides that are parallel and of equal length. By the Opposite Sides Parallel and Congruent Theorem, this quadrilateral is a parallelogram. Next, let's recall the Parallelogram Opposite Angles Theorem.

Parallelogram Opposite Angles Theorem
If a quadrilateral is a parallelogram then its opposite angles are congruent.

By using this theorem we can equate the measures of the angles from the diagram. 3x+25^(∘) = 2x + 50^(∘) Let's solve this equation for x.

3x+25^(∘) = 2x + 50^(∘)

SubEqn

LHS-2x=RHS-2x

x +25^(∘) = 50^(∘)

SubEqn

LHS-25^(∘)=RHS-25^(∘)

x = 25^(∘)

b We want to find the value of t based on the information regarding the perimeter and the following diagram.

Since the perimeter of a polygon is the sum of all of its side lengths, we can combine the information from the diagram with the given value of the perimeter. (5t+1)+(2t+5)+(3t-2)+t = 202 Let's solve the equation above for t.

(5t+1)+(2t+5)+(3t-2)+t = 202

AddSubTerms

Add and subtract terms

11t+4=202

SubEqn

LHS-4=RHS-4

11t = 198

DivEqn

.LHS /11.=.RHS /11.

t = 8

c Like in previous parts, let's first take a look at the given diagram.

We are told that the quadrilateral CARD is a rhombus. Next, let's recall the Rhombus Diagonals Theorem.

Rhombus Diagonals Theorem
A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Therefore, since CARD is a rhombus, its diagonals are perpendicular. This means that the measures from the diagram must equal 90^(∘). 4x-2^(∘) = 90^(∘) Let's solve the equation above for x.

4x-2^(∘) = 90^(∘)

AddEqn

LHS+2^(∘)=RHS+2^(∘)

4x = 92^(∘)

DivEqn

.LHS /4.=.RHS /4.

x = 23^(∘)

d Like in Parts A-C, let's first take a look at the given diagram.

It depicts a trapezoid with a pair of angles of equal measure near one of the bases. Let's recall the Converse Isosceles Trapezoid Base Angles Theorem.

Converse Isosceles Trapezoid Base Angles Theorem
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

By this theorem we know that our trapezoid is isosceles. This means that the legs of this trapezoid are congruent, which means that we can equate the expressions for their lengths. 13m - 9 = 7m+15 Let's solve the equation above for m.

13m - 9 = 7m+15

SubEqn

LHS-7m=RHS-7m

6m-9 = 15

AddEqn

LHS+9=RHS+9

6m = 24

DivEqn

.LHS /6.=.RHS /6.

m = 4

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