6. a) If \( (2 x-10)^{\circ} \) and \( (3 x+20)^{\circ} \) are a linear pair, find them. b) A nair of complementary angles are in the ratio \( 2: 3 \), find them.

To solve the problems, let's break them down one by one. ### Part a If \( (2x - 10)^\circ \) and \( (3x + 20)^\circ \) are a linear pair, it means that they are supplementary angles. Therefore, their sum is \( 180^\circ \). We can set up the equation: \[ (2x - 10) + (3x + 20) = 180 \] Now, simplify the equation: \[ 2x - 10 + 3x + 20 = 180 \] \[ 5x + 10 = 180 \] Next, subtract 10 from both sides: \[ 5x = 170 \] Now, divide by 5: \[ x = 34 \] Now, we can find the measures of the angles: 1. For \( (2x - 10)^\circ \): \[ 2(34) - 10 = 68 - 10 = 58^\circ \] 2. For \( (3x + 20)^\circ \): \[ 3(34) + 20 = 102 + 20 = 122^\circ \] Thus, the angles are \( 58^\circ \) and \( 122^\circ \). ### Part b Let the two complementary angles be \( 2k \) and \( 3k \) (since they are in the ratio \( 2:3 \)). Since they are complementary, their sum is \( 90^\circ \). We can set up the equation: \[ 2k + 3k = 90 \] Simplifying gives: \[ 5k = 90 \] Now, divide by 5: \[ k = 18 \] Now we can find the measures of the angles: 1. For the first angle \( 2k \): \[ 2(18) = 36^\circ \] 2. For the second angle \( 3k \): \[ 3(18) = 54^\circ \] Thus, the two complementary angles are \( 36^\circ \) and \( 54^\circ \). ### Summary of Answers a) The angles are \( 58^\circ \) and \( 122^\circ \). b) The complementary angles are \( 36^\circ \) and \( 54^\circ \).