There are four interior angle in a parallelogram and also the amount of the inner angles of a parallel is constantly 360°. The opposite angles of a parallelogram space equal and also the consecutive angles of a parallelogram space supplementary. Let united state read an ext about the properties of the angle of a parallel in detail.
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| 1. | Properties of angles of a Parallelogram |
| 2. | Theorems concerned Angles the a Parallelogram |
| 3. | FAQs on angles of a Parallelogram |
A parallelogram is a quadrilateral with equal and also parallel the contrary sides. There are some unique properties the a parallel that do it different from the various other quadrilaterals. Watch the following parallelogram come relate come its properties provided below:
Theorems regarded Angles that a Parallelogram
The theorems related to the angle of a parallelogram are beneficial to fix the problems related to a parallelogram. Two of the important theorems are given below:
The opposite angles of a parallelogram space equal.Consecutive angle of a parallelogram are supplementary.Let us learn about these two special theorems the a parallelogram in detail.
Opposite angles of a Parallelogram are Equal
Theorem: In a parallelogram, the contrary angles space equal.
Given: ABCD is a parallelogram, with 4 angles ∠A, ∠B, ∠C, ∠D respectively.
To Prove: ∠A =∠C and also ∠B=∠D
Proof: In the parallel ABCD, diagonal line AC is splitting the parallelogram right into two triangles. On comparing triangles ABC, and also ADC. Below we have:AC = AC (common sides)∠1 = ∠4 (alternate inner angles)∠2 = ∠3 (alternate internal angles)Thus, the 2 triangles room congruent, △ABC ≅ △ADCThis provides ∠B = ∠D by CPCT (corresponding components of congruent triangles).Similarly, we can show that ∠A =∠C.Hence proved, that opposite angle in any parallelogram are equal.
The converse the the above theorem says if the opposite angle of a quadrilateral room equal, then it is a parallelogram. Let us prove the same.
Given: ∠A =∠C and also ∠B=∠D in the quadrilateral ABCD.To Prove: ABCD is a parallelogram.Proof:The amount of every the 4 angles that this square is equal to 360°.= <∠A + ∠B + ∠C + ∠D = 360º>= 2(∠A + ∠B) = 360º (We have the right to substitute ∠C with ∠A and also ∠D through ∠B since it is given that ∠A =∠C and also ∠B =∠D)= ∠A + ∠B = 180º . This shows that the continuous angles space supplementary. Hence, it method that advertisement || BC. Similarly, we can present that abdominal || CD.Hence, ad || BC, and abdominal || CD.Therefore ABCD is a parallelogram.
Consecutive angles of a Parallelogram are Supplementary
The consecutive angle of a parallelogram space supplementary. Let us prove this property considering the adhering to given fact and using the same figure.
Given: ABCD is a parallelogram, with 4 angles ∠A, ∠B, ∠C, ∠D respectively.To prove: ∠A + ∠B = 180°, ∠C + ∠D = 180°.Proof: If advertisement is considered to it is in a transversal and abdominal || CD.According come the building of transversal, we understand that the inner angles ~ above the very same side of a transversal room supplementary.Therefore, ∠A + ∠D = 180°.Similarly,∠B + ∠C = 180°∠C + ∠D = 180°∠A + ∠B = 180°Therefore, the sum of the corresponding two nearby angles the a parallelogram is equal to 180°.Hence, the is showed that the consecutive angles of a parallelogram are supplementary.
Related posts on angles of a Parallelogram
Check the end the interesting articles given below that are pertained to the angle of a parallelogram.
Example 1: One edge of a parallelogram procedures 75°. Uncover the measure of its nearby angle and also the measure up of every the remaining angles the the parallelogram.
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Solution:
Given that one angle of a parallel = 75°Let the surrounding angle it is in xWe recognize that the continually (adjacent) angle of a parallelogram are supplementary.Therefore, 75° + x° = 180°x = 180° - 75° = 105°To uncover the measure up of all the four angles the a parallelogram we know that the opposite angles of a parallelogram room congruent.Hence, ∠1 = 75°, ∠2 = 105°, ∠3 = 75°, ∠4 = 105°
Example 2: The values of the opposite angle of a parallelogram are provided as follows: ∠1 = 75°, ∠3 = (x + 30)°, discover the worth of x.Given: ∠1 and also ∠3 are opposite angle of a parallelogram.
Solution:
Given: ∠1 = 75° and ∠3 = (x + 30)°We understand that the opposite angle of a parallelogram space equal.Therefore,(x + 30)° = 75°x = 75° - 30°x = 45°Hence, the worth of x is 45°.