Alternate Interior Angles Theorem Converse
Alternating Interior Angles Theorem
Alternate Interior Angles theorem states, if 2 parallel lines are cutting by a transversal, and then the pairs of alternate interior angles are congruent. Alternating interior angles are the angles formed on the opposite sides of the transversal. The alternate interior angles can bear witness whether the given lines are parallel or not. Permit us larn more than most the alternate interior angles theorem, the proof, and solve a few examples.
| 1. | What is the Alternate Interior Angles Theorem? |
| 2. | Definition of Alternate Interior Angles |
| 3. | Alternate Interior Angles Theorem and Proof |
| 4. | Alternating Interior Angles Theorem Converse |
| 5. | Co-interior Angles Theorem and Proof |
| six. | FAQs on Alternate Interior Angles Theorem |
What is the Alternate Interior Angles Theorem?
The alternate interior angles theorem states that if a transversal crosses the prepare of parallel lines, the alternate interior angles are congruent. In the figure given beneath, a set of parallel lines thousand and northward are intersected by the transversal and the post-obit pairs of alternate interior angles are formed: ∠one and ∠two, ∠iii and ∠4.
Since the given lines m and n are parallel, therefore the alternating interior angles will exist congruent. ∠1 = ∠2 and ∠3 = ∠four.
Interior angles on the same side of the transversal are called consecutive interior angles or co-interior angles in short. Co-interior angles are supplementary when the lines are parallel.
∠2 + ∠3 = 180° and ∠vi +∠7 = 180°
Corresponding angles are a pair of angles on the similar corners of each of two lines on the same side of the transversal line. Corresponding angles formed by 2 parallel lines and a transversal are equal. Corresponding angles in the to a higher place image are: ∠ii and ∠4, ∠i and ∠iii, ∠5 and ∠vii, and ∠6 and ∠8.
Definition of Alternate Interior Angles
When two parallel lines are crossed by a transversal, the pair of angles formed on the inner side of the parallel lines, but on the reverse sides of the transversal are called alternate interior angles. Alternating angles are of two types: Alternate interior angles and Alternating outside angles
- Alternating interior angles: Alternate interior angles are the angles formed when a transversal intersects two coplanar lines. They are on the inner side of the coplanar lines but are on the alternate reverse sides of the transversal.
- Alternate outside angles: Alternate outside angles are formed on the exterior of the coplanar lines but on the alternate opposite sides of the transversal.
According to the image below:
Alternate interior angles are: ∠iii and ∠6, ∠4 and ∠five.
Alternating outside angles are: ∠1 and ∠8, ∠ii and ∠vii.
For example: Let u.s. effort to spot alternate interior angles in the given figure. Recall the lines do not have to be ever parallel for alternate angles to be formed.
Alternating interior angles are: ∠3 and ∠half-dozen, ∠iv and ∠5
Alternating exterior angles are: ∠1 and ∠8, ∠2 and ∠vii
Backdrop of Alternate Interior Angles
Hither are a few properties of the alternate interior angles:
- The angles are coinciding.
- Two lines that never intersect, are equidistant, and are coplanar are chosen parallel lines. The symbol for parallel to is Ii.
- If nosotros accept two lines (they don't accept to be parallel) and have a tertiary line that crosses them, then the crossing line is said to exist a transversal.
- The sum of the angles formed on the same side of the transversal which are inside the two parallel lines is always equal to 180°.
- In the instance of non – parallel lines, alternate interior angles don't have any specific properties.
Finding Alternate Interior Angles
For finding alternate interior angles, we use the Z test. Brand a zig-zag line including the parallel lines as shown in the diagram. Here, p and q are alternate interior angles. Similarly, a and d are also alternate interior angles.
Alternate Interior Angles Theorem and Proof
Statement: The theorem states that if a transversal intersects parallel lines, the alternate interior angles are congruent.
Given: Line p II line q
To prove: ∠2= ∠7 and ∠3 = ∠6
Proof: Suppose p and q are 2 parallel lines and t is the transversal that intersects p and q.
We know that, if a transversal intersects whatever two parallel lines, the corresponding angles and vertically reverse angles are coinciding.
Therefore,
∠i = ∠three ………..(i) [Respective angles]
∠one = ∠6 ………..(two) [Vertically opposite angles]
From equations (i) and (ii), we become-
∠iii = ∠vi ..................[Alternate interior angles]
Similarly,
∠two = ∠7
Hence, it is proved.
Alternate Interior Angles Theorem Converse
The converse of alternate interior angles theorem states that if two lines are intersected past a transversal forming congruent alternate interior angles, then the lines are parallel. Thus according to the converse of alternate interior angles theorem, the beneath-given lines will be parallel if ∠D is 40° and ∠B is 140°. This is because their corresponding alternate interior angles are of the measure 40° and 140°.
In decision, the alternate interior angles theorem states that the alternating interior angles volition exist equal if the lines are parallel, whereas its converse states that lines will be parallel if the alternate interior angles are congruent.
Co-interior Angles Theorem and Proof
Co-interior angles are the ii interior angles that are on the same side of the transversal which makes information technology supplementary and sums up to 180 degrees. The co-interior angles resemble the shape C due to the placement of the angles merely the angles are not equal to each other. The other names for co-interior angles are consecutive interior angles or the same side interior angles. The image beneath shows the shape of co-interiors.
Theorem: If the transversal intersects the 2 parallel lines, each pair of co-interior angles sums up to 180 degrees (supplementary angles).
Proof:
Let usa consider the image given beneath:
In the figure, angles three and five are the co interior angles and angles iv and six are the co-interior angles.
To prove: ∠3 and ∠5 are supplementary and ∠iv and ∠6 are supplementary.
Given that, a and b are parallel to each other and t is the transversal.
By the definition of linear pair,
∠one and ∠3 class the linear pair.
Similarly, ∠2 and ∠4 grade the linear pair.
Past using the supplement postulate,
∠1 and ∠iii are supplementary
(i.e.) ∠1 + ∠three = 180
Similarly,
∠two and ∠4 are supplementary
(i.e.) ∠2 + ∠iv = 180
By using the corresponding angles theorem, nosotros can write
∠1 ≅∠5 and ∠2 ≅ ∠half dozen
Thus, by using the substitution property, we can say,
∠3 and ∠5 are supplementary and ∠iv and ∠six are supplementary.
Hence, the co-interior angle theorem (consecutive interior bending) is proved.
The converse of this theorem is if a transversal intersects two lines, such that the pair of co-interior angles are supplementary, so the 2 lines are parallel.
Related Topics
Listed below are a few interesting topics related to the alternate interior angles theorem, accept a look.
- Vertical Angles
- Alternate Angles
- Same Side Interior Angles
Examples on Alternate Interior Angles Theorem
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Instance 1: Cathy has been asked to find the pairs of alternate interior angles from the given diagram. Tin can yous assistance her?
Solution: The alternate interior angles prevarication on "alternate" sides of the transversal, and they must exist in the "interior" of the two parallel lines. The alternating interior angles are: ∠q and ∠z, ∠r and ∠a.
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Example two: Julia asks her students if they tin can detect the relationship between \(\bending q\) and \(\bending d\) if the given lines \(m\) and \(due north\) are parallel. Can you lot tell how are \(\angle q\) and \(\angle d\) related to each other?
Solution: The given lines are parallel and are intersected past a transversal.
According to the alternate interior angles theorem, the two coinciding alternating interior angles are produced. They are: ∠q and ∠d,∠c and ∠p.
Therefore, ∠q and ∠d are congruent alternate interior angles.
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Case 3: If the given two lines are parallel and are intersected by a transversal, what volition be the mensurate of ∠10 and ∠Y?
Solution: The given lines are parallel and co-ordinate to the alternate interior angles theorem, the given angle of measure 85° and ∠X are coinciding alternate interior angles.
Therefore, 85° = ∠X
Similarly, 95° and Y are congruent alternate interior angles. Therefore, 95° = ∠Y.
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Practice Questions on Alternating Interior Angles Theorem
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FAQs on Alternate Interior Angles Theorem
What is the Alternate Interior Angles Theorem?
The alternate interior angles theorem states that if a transversal crosses the set of parallel lines, the alternating interior angles are congruent. Two lines that never intersect, are equidistant, and are coplanar are chosen parallel lines. The symbol for parallel to is Two. If we have ii lines (they don't have to be parallel) and have a third line that crosses them, then the crossing line is said to be a transversal.
What is the Converse of the Alternating Interior Angles Theorem?
Co-ordinate to the converse of the alternate interior angles theorem, if a transversal intersects ii lines such that the alternate interior angles are equal, and so the two lines are said to be parallel.
Are Alternating Interior Angles Congruent?
Aye, alternating interior angles are congruent.
How is the Alternate Interior Angles Theorem and the Alternate Exterior Angles Theorem Akin?
The alternate interior angles theorem states that if 2 parallel lines are cutting past a transversal, then the alternate interior angles are congruent. The alternate exterior angles theorem states that if two parallel lines are cutting by a transversal, then the alternate exterior angles are congruent. Thus, the ii theorems are akin with respect to the congruent angles produced.
Alternate Interior Angles Theorem Converse,
Source: https://www.cuemath.com/geometry/alternate-interior-angles-theorem/
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