The problem requires us to use the given figure to write a proof for each of the listed theorems: Theorem 2.9, Theorem 2.10, Theorem 2.11, Theorem 2.12, and Theorem 2.13. The figure shows two lines, $l$ and $m$, that are perpendicular to each other, forming four angles labeled 1, 2, 3, and 4.

1. Problem Description

The problem requires us to use the given figure to write a proof for each of the listed theorems: Theorem 2.9, Theorem 2.10, Theorem 2.11, Theorem 2.12, and Theorem 2.

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3. The figure shows two lines, $l$ and $m$, that are perpendicular to each other, forming four angles labeled 1, 2, 3, and

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2. Solution Steps

To provide the proofs, we need to know the specific theorems 2.9, 2.10, 2.11, 2.12, and 2.

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3. Without that information, we can provide general statements relating to perpendicular lines. We know that perpendicular lines form right angles. In the figure, line $l$ is perpendicular to line $m$. This implies that angle 1, angle 2, angle 3, and angle 4 are all right angles.

By the definition of perpendicular lines:

$l \perp m$

Then:

$m\angle 1 = 90^{\circ}$

$m\angle 2 = 90^{\circ}$

$m\angle 3 = 90^{\circ}$

$m\angle 4 = 90^{\circ}$

Also, angles 1 and 3 are vertical angles, and angles 2 and 4 are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent.

$m\angle 1 = m\angle 3$

$m\angle 2 = m\angle 4$

Since all angles are right angles, all angles are congruent to each other.

$m\angle 1 = m\angle 2 = m\angle 3 = m\angle 4 = 90^{\circ}$

Also, angles 1 and 2, angles 2 and 3, angles 3 and 4, and angles 1 and 4 are adjacent angles. The adjacent angles form linear pairs.

$m\angle 1 + m\angle 2 = 180^{\circ}$

$m\angle 2 + m\angle 3 = 180^{\circ}$

$m\angle 3 + m\angle 4 = 180^{\circ}$

$m\angle 1 + m\angle 4 = 180^{\circ}$

Without knowing the specific statements of theorems 2.9 through 2.13, the proofs must be generalized, referencing perpendicular lines, right angles, and vertical angles.

3. Final Answer

Without the statements of theorems 2.9, 2.10, 2.11, 2.12 and 2.13, I can only make general observations about the given figure. Lines $l$ and $m$ are perpendicular, so angles 1, 2, 3, and 4 are all right angles and have measures of 90 degrees. Angles 1 and 3 are congruent vertical angles, as are angles 2 and

4. Any pair of adjacent angles in the diagram form a linear pair and are supplementary.