Question

13 draw, lm that bisects ∠tlc. if ∠tlc = 8x + 26 and ∠clm = 3x + 15, find: x =
m∠tlc =
m∠tlm =

Explanation:

Step1: Use angle - bisector property

Since $\overline{LM}$ bisects $\angle TLC$, then $\angle CLM=\frac{1}{2}\angle TLC$. So, $2\angle CLM = \angle TLC$.

Step2: Substitute the given angle expressions

We know that $\angle TLC = 8x + 26$ and $\angle CLM=3x + 15$. Substituting into $2\angle CLM=\angle TLC$, we get $2(3x + 15)=8x + 26$.

Step3: Expand the left - hand side

Expand $2(3x + 15)$ to get $6x+30$. So the equation becomes $6x + 30=8x + 26$.

Step4: Solve for $x$

Subtract $6x$ from both sides: $30 = 2x+26$. Then subtract 26 from both sides: $4 = 2x$. Divide both sides by 2, we find $x = 2$.

Step5: Find $\angle TLC$

Substitute $x = 2$ into the expression for $\angle TLC$: $\angle TLC=8x + 26=8\times2+26=16 + 26=42$.

Step6: Find $\angle TLM$

Since $\angle TLM=\angle CLM$ and $\angle CLM = 3x + 15$, substituting $x = 2$ gives $\angle CLM=3\times2+15=6 + 15=21$, so $\angle TLM = 21$.

Answer:

$x = 2$
$m\angle TLC=42$
$m\angle TLM = 21$