Question

1. which best describes ∠1 & ∠3?

adjacent angles complementary angles
linear pair vertical angles

1. which best describes ∠3 & ∠4?

complementary angles linear pair
supplementary angles vertical angles

1. in the figure below, y is between x & z, and xz = 30. what is the value of a?

x----a + 9----y----2a----z

1. in the diagram, y is between x & z, and xz = 45. what is the length of yz?

x----6a - 5----y----4a----z

1. solve for x.

i----2x + 27----j----2x + 31----k
|←----14----→|

1. solve for x.

g----2x - 17----h----6----i
|←----x - 2----→|

1. if m∠abc = 46°, what is the value of x?

(4x - 3)° (3x)°

1. if m∠abc = 44°, what is the value of x?

(5x - 12)° (3x)°

1. m∠tsa = 22° & m∠tsr = 103°. find m∠asr.
2. m∠fgh = 150° & m∠fge = 30°. find m∠egh.

Explanation:

Problem 3

Step1: Apply segment addition postulate

Since \( Y \) is between \( X \) and \( Z \), we have \( XZ = XY + YZ \). Given \( XZ = 30 \), \( XY = a + 9 \), and \( YZ = 2a \), so the equation is \( (a + 9) + 2a = 30 \).

Step2: Simplify and solve for \( a \)

Combine like terms: \( 3a + 9 = 30 \). Subtract 9 from both sides: \( 3a = 30 - 9 = 21 \). Divide both sides by 3: \( a = \frac{21}{3} = 7 \).

Step1: Apply segment addition postulate

Since \( Y \) is between \( X \) and \( Z \), \( XZ = XY + YZ \). Given \( XZ = 45 \), \( XY = 6a - 5 \), and \( YZ = 4a \), so \( (6a - 5) + 4a = 45 \).

Step2: Solve for \( a \)

Combine like terms: \( 10a - 5 = 45 \). Add 5 to both sides: \( 10a = 50 \). Divide by 10: \( a = 5 \).

Step3: Find length of \( YZ \)

Substitute \( a = 5 \) into \( YZ = 4a \), so \( YZ = 4 \times 5 = 20 \).

Step1: Apply segment addition postulate

\( IK = IJ + JK \). Given \( IK = 14 \), \( IJ = 2x + 27 \), \( JK = 2x + 31 \), so \( (2x + 27) + (2x + 31) = 14 \).

Step2: Solve for \( x \)

Combine like terms: \( 4x + 58 = 14 \). Subtract 58: \( 4x = 14 - 58 = -44 \). Divide by 4: \( x = -11 \).

Answer:

\( 7 \)

Problem 4