Answer:
1. x = 30
2. m∠A = 105°
3. x = 4
4. HA = 103
Step-by-step explanation:
An isosceles trapezoid is a quadrilateral with one pair of parallel sides and two non-parallel sides, where the non-parallel sides are of equal length.
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Question 1
In an isosceles trapezoid, opposite angles are supplementary (sum to 180°). Given that angle A is opposite angle H, we can express this relationship as:
m∠A + m∠H = 180°
Substitute m∠A = 100° and m∠H = (3x - 10)° into the equation and solve for x:
100° + (3x - 10)° = 180°
100 + 3x - 10 = 180
90 + 3x = 180
90 + 3x - 90 = 180 - 90
3x = 90
3x ÷ 3 = 90 ÷ 3
x = 30
Therefore, the value of x is 30.
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Question 2
In an isosceles trapezoid, the base angles are the same, and the opposite angles are supplementary (sum to 180°).
As angles A and H are opposite angles, their sum is 180°:
m∠A + m∠H = 180°
Since angles T and A are base angles, and m∠T = (111 - x)°, then it follows that m∠A = (111 - x)°. Given that m∠H = (10x + 15)°, we can substitute these expressions into the equation and solve for x:
(111 - x)° + (10x + 15)° = 180°
111 - x + 10x + 15 = 180
9x + 126 = 180
9x + 126 - 126 = 180 - 126
9x = 54
9x ÷ 9 = 54 ÷ 9
x = 6
Therefore, the value of x is 6.
To find the measure of angle A, substitute the found value of x into the angle expression:
m∠A = (111 - 6)°
m∠A = 105°
Therefore, the measure of angle A is 105°.
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Question 3
In an isosceles trapezoid, the diagonals are equal in length.
As MT and HA are the diagonals , and given that MT = 10x + 7 and HA = 8x + 15, we can determine the value of x by setting the expressions for the diagonals equal to each other and solving for x:
MT = HA
10x + 7 = 8x + 15
10x + 7 - 8x = 8x + 15 - 8x
2x + 7 = 15
2x + 7 - 7 = 15 - 7
2x = 8
2x ÷ 2 = 8 ÷ 2
x = 4
Therefore, the value of x is 4.
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Question 4
In an isosceles trapezoid, the diagonals are equal in length.
As MT and HA are the diagonals , and given that MT = 7x - 9 and HA = 6x + 7, we can determine the value of x by setting the expressions for the diagonals equal to each other and solving for x:
MT = HA
7x - 9 = 6x + 7
7x - 9 - 6x = 6x + 7 - 6x
x - 9 = 7
x - 9 + 9 = 7 + 9
x = 16
Therefore, the value of x is 16.
To find HA, substitute the value of x into the expression for HA:
HA = 6(16) + 7
HA = 96 + 7
HA = 103
So, the length of HA is 103 units.
