in a word processing document or on a separate piece of paper, use the guide to construct a two column proof…

in a word processing document or on a separate piece of paper, use the guide to construct a two column proof proving that triangle adc is congruent to triangle cba given that $\\angle d$ and $\\angle b$ are right angles and $dc$ is parallel to $ab$. upload the entire proof below.\n\ngiven:\n$\\angle d$ and $\\angle b$ are right angles\n$dc \\parallel ab$\n\nprove:\n$\\triangle adc \\cong \\triangle cba$\n\n| statement | reason |\n| :--- | :--- |\n| 1. $\\angle d$ and $\\angle b$ are right angles; $dc \\parallel ab$ | 1. given |\n| 2. | 2. if lines are parallel, then alternate interior angles are equal |\n| 3. | 3. |\n| 4. | 4. hypotenuse angle theorem |

in a word processing document or on a separate piece of paper, use the guide to construct a two column proof proving that triangle adc is congruent to triangle cba given that $\\angle d$ and $\\angle b$ are right angles and $dc$ is parallel to $ab$. upload the entire proof below.\n\ngiven:\n$\\angle d$ and $\\angle b$ are right angles\n$dc \\parallel ab$\n\nprove:\n$\\triangle adc \\cong \\triangle cba$\n\n| statement | reason |\n| :--- | :--- |\n| 1. $\\angle d$ and $\\angle b$ are right angles; $dc \\parallel ab$ | 1. given |\n| 2. | 2. if lines are parallel, then alternate interior angles are equal |\n| 3. | 3. |\n| 4. | 4. hypotenuse angle theorem |

Answer

Explanation:

Step1: Identify alternate interior angles

Since $DC \parallel AB$ and $AC$ is a transversal, the alternate interior angles $\angle DCA$ (labeled as 2) and $\angle CAB$ (labeled as 1) are congruent. $$\angle DCA \cong \angle CAB$$

Step2: Identify the shared side

Both triangles $\triangle ADC$ and $\triangle CBA$ share the side $AC$, which serves as the hypotenuse for both right triangles. $$AC \cong AC$$

Step3: State the reason for shared side

The property that a segment is congruent to itself is the Reflexive Property of Congruence. $$\text{Reflexive Property of Congruence}$$

Step4: Apply congruence theorem

With congruent hypotenuses ($AC$) and a pair of congruent acute angles ($\angle DCA \cong \angle CAB$), the triangles are congruent by the Hypotenuse-Angle (HA) Theorem. $$\triangle ADC \cong \triangle CBA$$

Answer:

STATEMENT REASON
1. $\angle D$ and $\angle B$ are right angles; $DC \parallel AB$ 1. Given
2. $\angle DCA \cong \angle CAB$ 2. If lines are parallel, then alternate interior angles are equal
3. $AC \cong AC$ 3. Reflexive Property of Congruence
4. $\triangle ADC \cong \triangle CBA$ 4. Hypotenuse Angle Theorem