There are four interior angles in a parallelogram and the sum of the inner angles that a parallel is constantly 360°. The opposite angle of a parallelogram room equal and also the consecutive angles of a parallelogram room supplementary. Let united state read an ext about the properties of the angle of a parallelogram in detail.
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|1.||Properties of angle of a Parallelogram|
|2.||Theorems regarded Angles of a Parallelogram|
|3.||FAQs on angles of a Parallelogram|
A parallelogram is a quadrilateral through equal and parallel opposite sides. There room some distinct properties that a parallel that do it different from the other quadrilaterals. Watch the following parallelogram come relate come its properties offered below:
Theorems related to Angles of a Parallelogram
The theorems related to the angle of a parallel are beneficial to settle the troubles related to a parallelogram. Two of the important theorems are provided below:The opposite angles of a parallelogram space equal.Consecutive angles of a parallelogram space supplementary.
Let united state learn around these two special theorems that a parallelogram in detail.
Opposite angle of a Parallelogram room Equal
Theorem: In a parallelogram, opposing angles space equal.
Given: ABCD is a parallelogram, with four angles ∠A, ∠B, ∠C, ∠D respectively.
To Prove: ∠A =∠C and also ∠B=∠D
Proof: In the parallel ABCD, diagonal line AC is separating the parallelogram into two triangles. On comparing triangles ABC, and also ADC. Right here we have:AC = AC (common sides)∠1 = ∠4 (alternate interior angles)∠2 = ∠3 (alternate internal angles)Thus, the 2 triangles are congruent, △ABC ≅ △ADCThis gives ∠B = ∠D by CPCT (corresponding components of congruent triangles).Similarly, we can show that ∠A =∠C.Hence proved, that opposite angles in any kind of parallelogram are equal.
The converse the the above theorem says if the opposite angles of a quadrilateral room equal, then it is a parallelogram. Let us prove the same.
Given: ∠A =∠C and ∠B=∠D in the square ABCD.To Prove: ABCD is a parallelogram.Proof:The sum of all the four angles the this quadrilateral is same to 360°.= <∠A + ∠B + ∠C + ∠D = 360º>= 2(∠A + ∠B) = 360º (We can substitute ∠C v ∠A and also ∠D with ∠B because it is provided that ∠A =∠C and also ∠B =∠D)= ∠A + ∠B = 180º . This mirrors that the continuous angles space supplementary. Hence, it way that ad || BC. Similarly, us can display that abdominal || CD.Hence, ad || BC, and abdominal || CD.Therefore ABCD is a parallelogram.
Consecutive angles of a Parallelogram room Supplementary
The consecutive angles of a parallelogram space supplementary. Let united state prove this building considering the adhering to given fact and using the very same figure.
Given: ABCD is a parallelogram, with four 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 property of transversal, we understand that the inner angles top top the exact 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 respective two nearby angles the a parallel is same to 180°.Hence, it is showed that the consecutive angles of a parallelogram space supplementary.
Related short articles on angles of a Parallelogram
Check the end the interesting articles given listed below that are pertained to the angle of a parallelogram.
Example 1: One edge of a parallelogram steps 75°. Discover the measure of its adjacent angle and the measure of all the remaining angles the the parallelogram.
See more: How Many Centimeters Are In 8 Inches To Centimeters Converter
Given the one angle of a parallelogram = 75°Let the adjacent angle be xWe know that the continually (adjacent) angle of a parallelogram room supplementary.Therefore, 75° + x° = 180°x = 180° - 75° = 105°To find the measure up of all the four angles that a parallel we understand that the opposite angles of a parallelogram space congruent.Hence, ∠1 = 75°, ∠2 = 105°, ∠3 = 75°, ∠4 = 105°
Example 2: The values of the opposite angles of a parallelogram are provided as follows: ∠1 = 75°, ∠3 = (x + 30)°, discover the worth of x.Given: ∠1 and ∠3 space opposite angle of a parallelogram.
Given: ∠1 = 75° and ∠3 = (x + 30)°We understand that the opposite angles of a parallelogram are equal.Therefore,(x + 30)° = 75°x = 75° - 30°x = 45°Hence, the worth of x is 45°.