Question

suppose you are given the following triangles with \\( \overline { a d } \cong \overline { p q } \\) and \\( \angle d \cong \angle q \\).
which additional information is needed to prove \\( \triangle a d j \cong \triangle p q r \\) by the asa triangle congruence theorem.
\\( \overline { d j } \cong \overline { q r } \\)
\\( \angle j \cong \angle r \\)
\\( \angle a \cong \angle p \\)
\\( \overline { j a } \cong \overline { r p } \\)
question 5 2 pts
suppose you cute off the angles in a diagram as shown above.
you lay the two pieces down, over top of \\( \angle 4 \\), so that they are adjacent to \\( \angle 3 \\), with their vertices at the same point so all three angles are adjacent to each other.
what theorem did you demonstrate?
base angles are congruent.
a straight angle has a measure of \\( 180 ^ { \circ } \\).
the transitive property of equality.
the sum of the remote interior angles equals the exterior angle.
corresponding angles have equal measure.

Explanation:

Question 1

Step1: Recall ASA congruence

ASA (Angle - Side - Angle) congruence theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
We are given \(AD\cong PQ\) and \(\angle D\cong\angle Q\).

Step2: Identify the needed angle

For \(\triangle ADJ\) and \(\triangle PQR\), the side \(AD\) (in \(\triangle ADJ\)) and \(PQ\) (in \(\triangle PQR\)) are the sides. The angles \(\angle D\) and \(\angle Q\) are given. The included angles for the sides \(AD\) and \(PQ\) are \(\angle A\) (in \(\triangle ADJ\)) and \(\angle P\) (in \(\triangle PQR\)) respectively.

Step1: Recall angle - sum property

When we cut off the angles of a triangle (\(\angle1,\angle2,\angle3\)) and place them adjacent to \(\angle4\) (an exterior angle of the triangle), we know that for a triangle, the exterior angle (\(\angle4\)) is equal to the sum of the two non - adjacent (remote) interior angles (\(\angle1\) and \(\angle2\)).

Answer:

\(\angle A\cong\angle P\)

Question 2