Question

question 4/25
refer to the figure shown below. consider the statements given in the first column and match these statements with their corresponding reasons in order to prove triangle sum theorem.
draw bd parallel to ac.
$mangle4 + mangle2 + mangle5=180^{circ}$
$angle1congangle4,angle3congangle5$
$mangle1 = mangle4,mangle3 = mangle5$
$mangle1 + mangle2 + mangle3 = 180^{circ}$
drag & drop the answer
triangle sum theorem
parallel postulate
alternate interior angles theorem
definition of congruent angles

Explanation:

Step1: Identify the first - step reason

Drawing a line parallel to a side of a triangle is based on the Parallel Postulate. So, "Draw BD parallel to AC" matches with "Parallel Postulate".

Step2: Identify the reason for the angle - sum on a straight - line

The sum of angles on a straight line is 180 degrees. For the angles $\angle4$, $\angle2$, and $\angle5$ which lie on the straight line BD, "$m\angle4 + m\angle2 + m\angle5=180^{\circ}$" matches with the fact that they are angles on a straight line (but among the given options, it is a pre - step for the proof).

Step3: Identify the reason for angle congruence

Since BD is parallel to AC, by the Alternate Interior Angles Theorem, $\angle1\cong\angle4$ and $\angle3\cong\angle5$. So, "$\angle1\cong\angle4,\angle3\cong\angle5$" matches with "Alternate Interior Angles Theorem".

Step4: Identify the reason for equal angle measures

The definition of congruent angles states that if two angles are congruent, their measures are equal. So, "$m\angle1 = m\angle4,m\angle3 = m\angle5$" matches with "Definition of congruent angles".

Step5: Identify the final step reason

By substituting $\angle4$ with $\angle1$ and $\angle5$ with $\angle3$ in the equation $m\angle4 + m\angle2 + m\angle5 = 180^{\circ}$, we get $m\angle1 + m\angle2 + m\angle3=180^{\circ}$, which is the Triangle Sum Theorem. So, "$m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$" matches with "Triangle Sum Theorem".

Answer:

Draw BD parallel to AC. - Parallel Postulate
$m\angle4 + m\angle2 + m\angle5 = 180^{\circ}$ - (Pre - step for proof, related to angles on a straight line)
$\angle1\cong\angle4,\angle3\cong\angle5$ - Alternate Interior Angles Theorem
$m\angle1 = m\angle4,m\angle3 = m\angle5$ - Definition of congruent angles
$m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$ - Triangle Sum Theorem