The image depicts a geometric problem involving a rectangle and its diagonal, with the task of proving that the sum of two angles equals 90 degrees. The solution involves applying the properties of parallel lines and triangles.
The image presents a geometric problem involving a rectangle ABCD, with a diagonal line segment AC dividing the rectangle into two right triangles, ACD and BAC. Given that angle DAB is a right angle and lines AB and CD are parallel, the task is to prove that the sum of angles 1 and 2 equals 90 degrees.
In this scenario, we can apply the properties of parallel lines cut by a transversal line to solve the problem. Since AB is parallel to CD and AC acts as a transversal line, it follows that alternate interior angles are congruent. Therefore, angle 1 (marked adjacent to point C) is congruent to angle DAB which is given as a right angle (90 degrees).
Furthermore, considering triangle ACD which has been formed by diagonal AC; since one of its angles (angle DAB) is already established as being 90 degrees and it’s known that the sum of all angles in any given triangle equals 180 degrees, we can calculate angle 2 (adjacent to point A). By subtracting the right-angle’s value from the total sum of triangle’s angles i.e., (180° - 90° = 90°), it becomes evident that the combined measure of angles CAD (angle 2) and DAC must be equal to 90 degrees.
Now having established these facts, let's focus on proving (m∠1 + m∠2 = 90°). Since (m∠1) is congruent to (m∠DAB) due to alternate interior angles being equal when lines are parallel and cut by a transversal; hence (m∠1 = m∠DAB = 90°). Also from above deduction in triangle ACD; (m∠DAC + m∠CAD = m∠2 = 90°).
So adding up both equations:
m∠1 + m∠2 = m ∠DAB + m ∠DAC + m ∠CAD
= (m ∠DAC + m ∠CAD) + m ∠DAB
![\[= (180 - \text{right-angle})+ \text{right-angle} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/cxapydebuh23yaqoosae51pveo8tr1vg88.png)
=180 °.
However this result contradicts our objective which was proving m∠1+m ∩2=9 °0.