Question

tuesday 8/26
each figure shows a triangle with one of its angle bisectors.

1. (mangle2 = 27x - 2) and (mangle1 = 25x + 2). find (x).
2. (mangle2 = 6x - 5) and (mangle1 = 5x + 1). find (x).
3. given (angle r) and (angle t) are complimentary. if (mangle r) is 135 what is the (mangle t)?
4. suppose (angle a) and (angle b) are supplementary angles, (mangle a=(3x + 5)^{circ}), and (mangle b=(2x - 10)^{circ}). solve for (x) and then find (mangle a) and (mangle b).

Explanation:

1.

Step1: Use angle - bisector property

Since an angle - bisector divides an angle into two equal angles, $m\angle1 = m\angle2$. So, $27x−2=25x + 2$.

Step2: Solve the equation for $x$

Subtract $25x$ from both sides: $27x-25x−2=25x-25x + 2$, which simplifies to $2x−2 = 2$. Then add 2 to both sides: $2x-2 + 2=2 + 2$, giving $2x=4$. Divide both sides by 2: $x = 2$.

Step1: Use angle - bisector property

Because an angle - bisector makes $m\angle1 = m\angle2$, we have the equation $6x−5=5x + 1$.

Step2: Solve the equation for $x$

Subtract $5x$ from both sides: $6x-5x−5=5x-5x + 1$, which simplifies to $x−5 = 1$. Then add 5 to both sides: $x-5 + 5=1 + 5$, so $x = 6$.

Step1: Recall the definition of complementary angles

Complementary angles add up to $90^{\circ}$. But $m\angle R=135^{\circ}$, and since $135^{\circ}>90^{\circ}$, there is an error in the problem statement as an angle in a complementary - angle pair cannot be greater than $90^{\circ}$. If we assume it was a mis - type and they are supplementary (add up to $180^{\circ}$), then $m\angle T=180^{\circ}-m\angle R$.

Step2: Calculate $m\angle T$

$m\angle T=180 - 135=45^{\circ}$

Answer:

$x = 2$

2.