If PR bisects ∠SRT and U is the midpoint of RT.

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:

The complete question is:

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

The picture of the question in the attached figure

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

Classify

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

Classify

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

Classify

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

Classify

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)

Classify

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

Classify

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

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for