• Opposite angles in parallelograms are congruent.
• Two adjacent angles adds up to 180°, they are supplementary angles.
[tex] \boxed { \begin{array}{} \tt m\angle ESY+m\angle AYS=180° \\ \tt m\angle AYS=180°-55°\\ \tt m\angle AYS=125° \end{array}}[/tex]
[tex]\boxed { \begin{array}{} \tt m\angle M+m\angle O=180° \\ \tt 4x - 9 + 2x + 3 = 180 \\ \tt6x - 6 = 180 \\ \tt6x = 180 + 6 \\ \tt 6x = 186 \\ \tt x = 31 \end{array}}[/tex]
• Since opposite angles in parallelograms are congruent, then the measure of the angle E is the same with measure of the angle O.
[tex]\boxed { \begin{array}{} \tt m \angle O \cong m\angle E\\ \tt m \angle E = 2x + 3 \\ \tt m \angle E = 2(31)+ 3 \\ \tt m \angle E = 62+ 3 \\\tt m \angle E = 65 \degree \end{array}}[/tex]
• Since opposite angles in parallelograms are congruent, then the of the angle M is the same with measure of the angle R.
[tex]\boxed { \begin{array}{} \tt m \angle M \cong m\angle R\\ \tt m \angle R = 4x-9 \\ \tt m \angle R = 4(31) - 9\\ \tt m \angle R = 124 - 9\\\tt m \angle R = 115 \degree \end{array}}[/tex]
• Two adjacent angles adds up to 180°, they are supplementary angles.
[tex] \boxed { \begin{array}{} \tt m\angle I+m\angle L=180° \\ \tt m\angle L=180°-50°\\ \tt m\angle L=130° \end{array}}[/tex]
• Opposite angles in parallelograms are congruent.
• Opposite angles in parallelograms are congruent.
[tex]\boxed { \begin{array}{} \tt m \angle S \cong m\angle M\\ \tt8x - 6 = 6x + 20 \\ \tt 2x = 20 + 6 \\ \tt 2x = 26 \\ \tt x = 13 \end{array}}[/tex]
• Substitute the value of x to the expression, 8x - 6.
[tex]\boxed { \begin{array}{} \tt m \angle S = 8x - 6 \\ \tt m \angle S = 8(13) - 6 \\\tt m \angle S = 104- 6 \\ \tt m \angle S = 98 \degree \end{array}}[/tex]
• Two adjacent angles adds up to 180°, they are supplementary angles.
[tex]\boxed { \begin{array}{} \tt m\angle S+m\angle E=180° \\ \tt m\angle E=180°-98°\\ \tt m\angle E=82° \end{array}}[/tex]
Angle properties of parallelogram:
SOLUTION:
1. What is the measure of ∠Y in parallelogram EASY?
Observe that ∠E is the opposite angle of ∠Y.
Opposite angles are congruent.
The measure of ∠E is 105° (given), so ∠Y ≅ 105°
Therefore, C. 105° is the correct answer.
2. In a parallelogram EASY if m∠ESY = 55°, what is m∠AYS?
Since m∠ESY and m∠AYS are consecutive angles, they are supplementary.
So we need to equate their sum to 180°
[tex]\sf m\angle ESY + m\angle AYS = 180^{\circ}[/tex]
Given that m∠ESY = 55°, plugging in, we have...
[tex]\sf 55^{\circ} + m\angle AYS = 180^{\circ}[/tex]
[tex]\sf 55^{\circ} + m\angle AYS - 55^{\circ} = 180^{\circ} - 55^{\circ}[/tex]
[tex]\sf m\angle AYS = \boxed{\sf 125^{\circ}}[/tex]
Therefore, C. 125° is the correct answer.
3. In a parallelogram MORE, m∠M = (4x - 19)° and m∠O = (2x + 3)°, what is the value of x?
Since ∠M and ∠O are consecutive angles, they are supplementary
So,
[tex]\sf (4x - 9)^{\circ} + (2x + 3)^{\circ} = 180^{\circ}[/tex]
[tex]\sf (6x - 6)^{\circ} = 180^{\circ}[/tex]
[tex]\sf 6x = 180 + 6[/tex]
[tex]\sf 6x = 186[/tex]
[tex]\sf \frac{6x}{6} = \frac{186}{6}[/tex]
[tex]\sf x = \boxed{\sf 31}[/tex]
Therefore, B. 31 is the correct answer.
4. In parallelogram MORE, what is the measure of angle E?
Observe that ∠E and ∠O are opposite angles, thus they are congruent.
It follows,
[tex]\sf m\angle E = (2x + 3)^{\circ}[/tex]
[tex]\sf m \angle E = (2(31) + 3)^{\circ}[/tex]
[tex]\sf m\angle E = \boxed{\sf 65^{\circ}}[/tex]
Therefore, A. 65° is the correct answer.
5. In parallelogram MORE, what is the measure of angle R?
Observe that ∠M and ∠R are opposite angles, thus they are congruent.
It follows,
[tex]\sf m\angle R = (4x-9)^{\circ}[/tex]
[tex]\sf m\angle R = (4(31) - 9)^{\circ}[/tex]
[tex]\sf m\angle R = \boxed{\sf 115^{\circ}}[/tex]
Therefore, C. 115° is the correct answer.
6. What is the measure of ∠L in parallelogram LIVE?
Since ∠L and ∠I are consecutive angles, they are supplementary
Given: ∠I = 50°.
So,
[tex]\sf m\angle L + 50^{\circ} = 180^{\circ}[/tex]
[tex]\sf m\angle L + 50^{\circ} - 50^{\circ} = 180^{\circ} - 50^{\circ}[/tex]
[tex]\sf m\angle L = \boxed{\sf 130^{\circ}}[/tex]
Therefore, D. 130° is the correct answer.
7. In a parallelogram LIVE, if m∠EVI = 115°, what is m∠ELI?
Observe that ∠EVI and ∠ELI are opposite angles, thus the angles are congruent.
Since m∠EVI = 115°, m∠ELI ≅ 115°
Therefore, B. 115° is the correct answer.
8. In a parallelogram SOME, m∠S = (8x - 6)° and m∠M = (6x + 20)° what is the value of x?
Observe that ∠S and ∠M are opposite angles, thus they are congruent.
It follows,
[tex]\sf (8x - 6)^{\circ} = (6x + 20)^{\circ}[/tex]
[tex]\sf 8x - 6x = 20 + 6[/tex]
[tex]\sf 2x = 26[/tex]
[tex]\sf \frac{2x}{2} = \frac{26}{2}[/tex]
[tex]\sf x = \boxed{\sf 13}[/tex]
Therefore, C. 13 is the correct answer.
9. In parallelogram SOME. what is the measure of angle S?
[tex]\sf m \angle S = (8x - 6)^{\circ}[/tex]
[tex]\sf m \angle S = (8(13)-6)^{\circ}[/tex]
[tex]\sf m \angle S = \boxed{\sf 98^{\circ}}[/tex]
Therefore, B. 98° is the correct answer.
10. In parallelogram SOME. what is the measure of angle E?
Observe that ∠S and ∠E are consecutive angles, thus they are supplementary.
It follows,
[tex]\sf 98^{\circ} + m\angle E = 180^{\circ}[/tex]
[tex]\sf 98 + m \angle E - 98^{\circ} = 180^{\circ} - 98^{\circ}[/tex]
[tex]\sf m \angle E = \boxed{\sf 82^{\circ}}[/tex]
Therefore, A. 82° is the correct answer.
ANSWERS:
1. C. 105°
2. C. 125°
3. B. 31
4. A. 65°
5. C. 115°
6. D. 130°
7. B. 115°
8. C. 13
9. B. 98°
10. A. 82°
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