Question

find the values of the variables and the measures of the angles.
29.
30.
31.
32.

Explanation:

Step1: Use angle - sum property of a triangle for problem 29

The sum of interior angles of a triangle is $180^{\circ}$. So, $(2x - 9)+(2x + 4)+x=180$.
Combining like - terms: $2x+2x + x-9 + 4=180$, which simplifies to $5x-5 = 180$.
Adding 5 to both sides: $5x=180 + 5=185$.
Dividing both sides by 5: $x=\frac{185}{5}=37$.

Step2: Find the angle measures for problem 29

Angle $P=(2x - 9)^{\circ}=(2\times37 - 9)^{\circ}=(74 - 9)^{\circ}=65^{\circ}$.
Angle $Q=(2x + 4)^{\circ}=(2\times37+4)^{\circ}=(74 + 4)^{\circ}=78^{\circ}$.
Angle $R=x^{\circ}=37^{\circ}$.

Step3: Use angle - sum property of a triangle for problem 30

Since the triangle has a right - angle, the sum of the two non - right angles is $90^{\circ}$. So, $(4x + 7)+(8x - 1)=90$.
Combining like - terms: $4x+8x+7 - 1=90$, which simplifies to $12x + 6=90$.
Subtracting 6 from both sides: $12x=90 - 6=84$.
Dividing both sides by 12: $x=\frac{84}{12}=7$.

Step4: Find the angle measures for problem 30

Angle $A=(4x + 7)^{\circ}=(4\times7 + 7)^{\circ}=(28 + 7)^{\circ}=35^{\circ}$.
Angle $B=(8x - 1)^{\circ}=(8\times7 - 1)^{\circ}=(56 - 1)^{\circ}=55^{\circ}$.

Step5: Use angle relationships for problem 31

In right - triangle $FGH$, $a = 90^{\circ}-55^{\circ}=35^{\circ}$ (complementary angles).
$b = 90^{\circ}-32^{\circ}=58^{\circ}$ (complementary angles).
In triangle $EFH$, $d=180^{\circ}-(32^{\circ}+55^{\circ})=93^{\circ}$ (angle - sum property of a triangle).
$e = 180^{\circ}-93^{\circ}=87^{\circ}$ (linear pair).

Step6: Use angle relationships for problem 32

In right - triangle $ABD$, $y=90^{\circ}-54^{\circ}=36^{\circ}$ (complementary angles).
In right - triangle $BCD$, $x=90^{\circ}-52^{\circ}=38^{\circ}$ (complementary angles).
$z = 90^{\circ}$ (right - angle).

Answer:

Problem 29: $x = 37$, Angle $P = 65^{\circ}$, Angle $Q = 78^{\circ}$, Angle $R = 37^{\circ}$
Problem 30: $x = 7$, Angle $A = 35^{\circ}$, Angle $B = 55^{\circ}$
Problem 31: $a = 35^{\circ}$, $b = 58^{\circ}$, $d = 93^{\circ}$, $e = 87^{\circ}$
Problem 32: $x = 38$, $y = 36$, $z = 90$