7.05 Proving triangles congruent
Interactive practice questions
In the diagram, $MQ$MQ is perpendicular to $PR$PR, and $\Delta MPR$ΔMPR is isosceles.
An isosceles triangle with vertex $M$M and base $RP$RP is depicted. Sides $MR$MR and $MP$MP are marked with single hash marks indicating congruence. A perpendicular line from vertex $M$M to the base is drawn, intersecting the base at point $Q$Q, forming a right angle, which is denoted by a small square at the intersection.
What is the measure of $\angle MQR$∠MQR?
From the information given, we know $\triangle MPQ$△MPQ and $\triangle MRQ$△MRQ are congruent because:
they have three pairs of congruent sides.
they have two pairs of congruent sides and the non-included angle is congruent.
they have two pairs of congruent sides and the included angle is congruent.
they have three pairs of congruent angles.
they are both right triangles with hypotenuse and one leg the same length.
Which congruence test matches our argument from the previous part?
Side-angle-side congruence (SAS)
Side-side-side congruence (SSS)
Angle-side-angle congruence (ASA)
Angle-angle-side congruence (AAS)
Hypotenuse-leg congruence (HL)
This two-column proof shows that $\Delta ABC\cong\Delta XYZ$ΔABC≅ΔXYZ in the attached diagram, but it is incomplete.
This two-column proof shows that $\Delta ABC\cong\Delta XYZ$ΔABC≅ΔXYZ , but it is incomplete.
This two-column proof shows that $\Delta PQR\cong\Delta RSP$ΔPQR≅ΔRSP in the attached diagram, but it is incomplete.