Question

given: $-3(2x + 7) = -29 - 4x$; prove: $x = 4$
statements reasons

1. given: $\overrightarrow{km}$ bisects $\angle jkn$, $\overrightarrow{kn}$ bisects $\angle mkl$

prove: $\angle jkm \cong \angle nkl$
statements reasons

1. $\overrightarrow{km}$ bisects $\angle jkn$ 1.
2. $\angle jkm \cong \angle mkn$ 2.
3. $\overrightarrow{kn}$ bisects $\angle mkl$ 3.
4. $\angle mkn \cong \angle nkl$ 4.
5. $\angle jkm \cong \angle nkl$ 5.

1. given: $\angle 1$ and $\angle 2$ form a linear pair, $\angle 2 \cong \angle 4$

prove: $\angle 1$ and $\angle 3$ are supplementary
statements reasons

1. $\angle 1$ and $\angle 2$ form a linear pair 1.
2. $\angle 1$ and $\angle 2$ are supplementary 2.
3. $m\angle 1 + m\angle 2 = 180^\circ$ 3.
4. $\angle 2 \cong \angle 4$ 4.
5. $\angle 3 \cong \angle 4$ 5.
6. $\angle 2 \cong \angle 3$ 6.
7. $m\angle 2 = m\angle 3$ 7.
8. $m\angle 1 + m\angle 3 = 180^\circ$ 8.
9. $\angle 1$ and $\angle 3$ are supplementary 9.

© gina wilson (all things algebra®, llc), 2014-2020

Explanation:

First Problem: Given: $-3(2x + 7) = -29 - 4x$; Prove: $x = 4$

Step1: Expand left-hand side

$-3(2x + 7) = -6x - 21$
Equation becomes: $-6x - 21 = -29 - 4x$

Step2: Add $6x$ to both sides

$-6x - 21 + 6x = -29 - 4x + 6x$
Simplify: $-21 = -29 + 2x$

Step3: Add 29 to both sides

$-21 + 29 = -29 + 2x + 29$
Simplify: $8 = 2x$

Step4: Divide by 2

$\frac{8}{2} = \frac{2x}{2}$
Simplify: $x = 4$

1. The first statement is given directly in the problem.
2. A bisector splits an angle into two congruent angles, so $\overrightarrow{KM}$ bisecting $\angle JKN$ creates two equal angles.
3. The second bisector statement is given in the problem.
4. A bisector splits an angle into two congruent angles, so $\overrightarrow{KN}$ bisecting $\angle MKL$ creates two equal angles.
5. By the transitive property of congruence, if $\angle JKM \cong \angle MKN$ and $\angle MKN \cong \angle NKL$, then the two outer angles are congruent.

1. The first statement is given directly in the problem.
2. Linear pairs of angles are always supplementary by definition.
3. Supplementary angles have a sum of $180^\circ$.
4. The second congruence statement is given directly in the problem.
5. $\angle 3$ and $\angle 4$ are vertical angles, which are congruent.
6. By the transitive property, if $\angle 2 \cong \angle 4$ and $\angle 3 \cong \angle 4$, then $\angle 2 \cong \angle 3$.
7. Congruent angles have equal measures.
8. Substitute $m\angle 2$ with $m\angle 3$ in the supplementary angle sum.
9. Angles with measures that add to $180^\circ$ are supplementary by definition.

Answer:

$x = 4$

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Problem 24: Given: $\overrightarrow{KM}$ bisects $\angle JKN$, $\overrightarrow{KN}$ bisects $\angle MKL$; Prove: $\angle JKM \cong \angle NKL$