Answer:
Part 1) Triangle UQR Is an obtuse isosceles triangle
Part 2) Triangle RST is a equilateral triangle
Part 3) Triangle SRQ is a right scalene triangle
Part 4) Triangle PRT Is an obtuse isosceles triangle
Part 5) Triangle TQU is a equilateral triangle
Part 6) Triangle PQT is a right scalene triangle
Step-by-step explanation:
<u><em>The complete question is:</em></u>
If PR bisects ∠SRT and U is the midpoint of RT, classify each triangle by its angles and sides
Triangle UQR
Triangle RST
Triangle SRQ
Triangle PRT
Triangle TQU
Triangle PQT
<u><em>The picture of the question in the attached figure</em></u>
Part 1) Triangle UQR
we know that
PR bisects ∠SRT
That means
we have
---> is given
so
Remember that the sum of the interior angles in any triangle must be equal to 180 degrees
so
In the triangle UQR
so
Triangle UQR is a
Remember that
An isosceles triangle has two equal sides and two equal interior angles
so
<u>Classify</u>
By its angles is an obtuse triangle (has an interior angle greater than 90 degrees)
By its sides is an isosceles triangle (has two equal sides)
therefore
Triangle UQR Is an obtuse isosceles triangle
Part 2) Triangle RST
we know that
---> because U is the midpoint of RT
----> because PR bisects ∠SRT
so
<u><em>Classify</em></u>
By its angles is an acute triangle ( interior angles less than 90 degrees)
By its sides is an equilateral triangle (has three equal sides)
therefore
Triangle RST is a equilateral triangle
Note: An equilateral triangle is subtended to be an acute triangle, because the measure of the interior angles is always 60 degrees
Part 3) Triangle SRQ
we know that
Triangle SRQ is a
<u><em>Classify</em></u>
By its angles is a right triangle (has an interior angle equal to 90 degrees)
By its sides is a scalene triangle (has three different sides)
Note: If any triangle has three different interior angles, then the triangle has three different length sides
therefore
Triangle SRQ is a right scalene triangle
Part 4) Triangle PRT
Find the length side QR
Applying the Pythagorean Theorem in the right triangle SRQ
Find the length side PQ
Applying the Pythagorean Theorem in the right triangle PQT
The length sides of triangle PRT are
The interior angle of triangle PRT are
<u>Classify</u>
By its angles is an obtuse triangle (has an interior angle greater than 90 degrees)
By its sides is an isosceles triangle (has two equal sides)
therefore
Triangle PRT Is an obtuse isosceles triangle
Part 5) Triangle TQU
Find the measure of angle TUQ
we know that
----> by supplementary angles (form a linear pair)
Find the measure of angle TQU
we know that
----> by complementary angles
----> see part 1)
so
The interior angle of triangle TQU are
Note: If a triangle has three equal interior angles, then the triangle has three equal sides (QT=QU=TU=8 m)
<u><em>Classify</em></u>
Triangle TQU is a equilateral triangle
Note: An equilateral triangle is subtended to be an acute triangle, because the measure of the interior angles is always 60 degrees
Part 6) Triangle PQT
we know that
Triangle PQT and Triangle RQS are congruent by SSS postulate
so
The interior angle of triangle PQT are
The length sides of triangle PQT are
<u><em>Classify</em></u>
By its angles is a right triangle (has an interior angle equal to 90 degrees)
By its sides is a scalene triangle (has three different sides)
Note: If any triangle has three different interior angles, then the triangle has three different length sides
therefore
Triangle PQT is a right scalene triangle