Math  /  Geometry

Question3. 34\angle 3 \equiv \angle 4
3. verticle <<
4. 42\angle 4 \geqslant \angle 2
5. <12<1 \cong \angle 2 4 corresponding << are ¥¥
5. Transitive
5. Transitive
2.  Given:  Prove:  ald m,<1<2\begin{array}{l}\text { Given: } \\ \text { Prove: } \\ \text { ald }\end{array} \mathrm{m},<1 \cong<2 \begin{tabular}{l|l} Statements & Reaspns \\ \hline 2. 11 m,L13211 \mathrm{~m}, L 13 \angle 2 & Given \\
2. Given & \end{tabular}

Studdy Solution

STEP 1

1. The lines l l and m m are parallel.
2. A transversal intersects the parallel lines l l and m m .
3. The angles formed by the intersection are labeled as 1 \angle 1 , 2 \angle 2 , 3 \angle 3 , and 4 \angle 4 .
4. The goal is to prove that 12 \angle 1 \cong \angle 2 .
5. The properties of vertical angles, corresponding angles, and alternate interior angles will be used.

STEP 2

1. Identify vertical angles and their relationships.
2. Identify corresponding angles and their relationships.
3. Apply the transitive property to establish the required congruence.

STEP 3

Identify the vertical angle pairs. Since 3 \angle 3 and 4 \angle 4 are vertical angles, we have: 34 \angle 3 \equiv \angle 4

STEP 4

Identify the corresponding angles. Since lm l \parallel m and they are intersected by a transversal, the corresponding angles are congruent. Therefore: 13 \angle 1 \cong \angle 3

STEP 5

Identify the alternate interior angles. Since lm l \parallel m , the alternate interior angles are congruent. Therefore: 42 \angle 4 \cong \angle 2

STEP 6

Apply the transitive property of angle congruence. From the previous steps, we have: 13and34and42 \angle 1 \cong \angle 3 \quad \text{and} \quad \angle 3 \equiv \angle 4 \quad \text{and} \quad \angle 4 \cong \angle 2
By the transitive property, if 13 \angle 1 \cong \angle 3 and 34 \angle 3 \equiv \angle 4 , then 14 \angle 1 \cong \angle 4 .
Then, if 42 \angle 4 \cong \angle 2 , we can conclude: 12 \angle 1 \cong \angle 2

STEP 7

Restate the conclusion clearly. Given that 13 \angle 1 \cong \angle 3 , 34 \angle 3 \equiv \angle 4 , and 42 \angle 4 \cong \angle 2 , by transitivity: 12 \angle 1 \cong \angle 2
Solution: We have successfully proven that 12 \angle 1 \cong \angle 2 given that lm l \parallel m .

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