25. mT = 251°
26. BD = 2x + 25
27. The value of WX is √113
28. The measure of angle QRS = 153°
29. Quadrilateral PORS is a parallelogram.
25. To find the measure of angle T, we can use the following equation:
mT = 360° - (92° + 17°)
mT = 360° - 109°
mT = 251°
26. To find the length of BD, we can use the following equation:
BD = AC - 8
BD = 2x + 33 - 8
BD = 2x + 25
27. we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs).
Steps to solve:
Identify the right triangle. In this case, triangle WXS is a right triangle, with WX being the hypotenuse.
Find the squares of the legs. We are given that WS = 7 and SX = 8. Therefore, WS^2 = 7^2 = 49 and SX^2 = 8^2 = 64.
Apply the Pythagorean theorem. WX^2 = WS^2 + SX^2, so WX^2 = 49 + 64 = 113.
Take the square root of both sides to find WX. WX = √113.
Therefore, the value of WX is √113.
28. Since QRST is a kite, we know that mLQRS = mLTS.
We are given that mQRS = 11x - 6.
To find mLTS, we can use the fact that the sum of the angles in a quadrilateral is 360°.
Therefore, we can write the following equation:
mLQRS + mLTS + mRST + mSQP = 360°
Substituting mLQRS = 11x - 6 and mLTS = mLQRS, we get:
(11x - 6) + (11x - 6) + mRST + mSQP = 360°
22x - 12 + mRST + mSQP = 360°
We are also given that mRST + mSQP = 57°. Substituting this into the previous equation, we get:
22x - 12 + 57° = 360°
22x = 315°
x = 15°
Therefore, mLQRS = mLTS = 11(15°) - 6 = 159° - 6 = 153°.
29. To determine if quadrilateral PORS is a parallelogram, we can use the following properties of parallelograms:
Opposite sides are equal in length.
Opposite sides are parallel.
We can use the distance formula to calculate the lengths of the sides of PORS:
PS = √((-6 - (-1))^2 + (4 - 0)^2)
PS = √(25 + 16)
PS = √41
OR = √((-1 - (-5))^2 + (0 - (-3))^2)
OR = √(16 + 9)
OR = √25
RS = √((-5 - (-1))^2 + (-3 - 0)^2)
RS = √(16 + 9)
RS = √25
PO = √((-6 - (-2))^2 + (4 - 7)^2)
PO = √(16 + 9)
PO = √25
We can see that the opposite sides of PORS are equal in length.
To determine if the opposite sides of PORS are parallel, we can use the slope formula:
Slope of PS = (4 - 0) / (-6 - (-1)) = 4 / -5 = -4/5
Slope of OR = (0 - (-3)) / (-1 - (-5)) = 3 / 4
The slopes of PS and OR are negative inverses of each other, so the lines are parallel.
Therefore, quadrilateral PORS is a parallelogram.