From the Following Figure, How Can You Conclude That Lines L and M Are Parallel?
Sequent Interior Angles
Consecutive interior angles are formed on the inner sides of the transversal and are also known as co-interior angles or same-side interior angles. When a transversal crosses any ii parallel lines, it forms many angles like alternate interior angles, corresponding angles, alternate exterior angles, consecutive interior angles. Let united states learn more well-nigh consecutive interior angles on this folio.
| 1. | What are Sequent Interior Angles? |
| 2. | Sequent Interior Angles Theorem |
| 3. | Converse of Sequent Interior Angles Theorem |
| iv. | FAQs on Consecutive Interior Angles |
What are Consecutive Interior Angles?
Sequent interior angles are divers as the pair of non-adjacent interior angles that lie on the same side of the transversal. The give-and-take 'sequent' refers to things that appear next to each other. Sequent interior angles are located next to each other on the internal side of a transversal. Find the post-obit figure and the properties of sequent interior angles to identify them.
- Consecutive interior angles accept different vertices.
- They lie between 2 lines.
- They are on the same side of the transversal.
- They share a common side.
In the figure given higher up, L1 and L2 are parallel lines and T is the transversal. By the consecutive interior angles definition, the pairs of consecutive interior angles in the effigy are:
- ∠1 and ∠4
- ∠2 and ∠three
Angles Formed by a Transversal
When a transversal crosses a pair of parallel lines, many pairs of angles are formed other than consecutive interior angles. They are respective angles, alternate interior angles, and alternate outside angles. Observe the following effigy and relate to the various pairs of angles and their properties given in the table.
The following table lists the properties of all the types of angles formed when a transversal crosses ii parallel lines. Refer to the figure given above to relate to the angles.
| Types of Angles | Properties | Name of the Angles in the Effigy |
|---|---|---|
| Respective Angles | Corresponding angles are those angles that:
When a transversal intersects two parallel lines, the respective angles formed are always equal. | In the above figure, ∠i & ∠5, ∠2 &∠6, ∠4 & ∠8, ∠3 & ∠seven are all pairs of respective angles. |
| Alternating Interior Angles | Alternate interior angles are those angles that:
When a transversal intersects two parallel lines, the alternate interior angles formed are always equal. | In the above figure,∠4 & ∠vi and ∠3 & ∠five are pairs of alternate interior angles. |
| Alternate Outside Angles | Alternate outside angles are those angles that:
When a transversal intersects 2 parallel lines, the alternating outside angles formed are e'er equal. | In the above effigy, ∠one & ∠vii and ∠2 & ∠viii are pairs of alternate exterior angles. |
| Consecutive Interior Angles | Sequent interior angles are those angles that:
When a transversal intersects two parallel lines, the sequent interior angles are e'er supplementary. | In the in a higher place figure, ∠iv & ∠v and ∠three & ∠vi are pairs of consecutive interior angles. |
Consecutive Interior Angle Theorem
The relation between the consecutive interior angles is determined by the sequent interior angle theorem. The 'consecutive interior bending theorem' states that if a transversal intersects two parallel lines, each pair of consecutive interior angles is supplementary, that is, the sum of the consecutive interior angles is 180°.
Proof of Consecutive Interior Angle Theorem
Observe the following figure to sympathize the Consecutive Interior Angle Theorem.
It is given that the 2 lines L1 and Ltwo are parallel, and T is the transversal. Since 50i // L2, it tin can be said that:
- ∠1 = ∠5 (corresponding angles with // lines) ----- (Equation ane)
- ∠1 + ∠4 = 180° (Linear pair of angles are supplementary) ---- (Equation 2)
- Substituting ∠1 every bit ∠v in Equation (2), we get, ∠v + ∠4 = 180°
- Similarly, we tin bear witness that, ∠iii +∠6 = 180°.
- ∠ii = ∠half-dozen (corresponding angles with // lines) ----- (Equation 3)
- ∠2 + ∠3 = 180° (Linear pair of angles are supplementary) ---- (Equation iv)
- Substituting ∠2 as ∠half dozen in Equation (iv), we get, ∠6 + ∠iii = 180°
- So, it can be seen that ∠4 + ∠5 = 180°; and ∠three + ∠6 = 180°
Therefore, it is proved that consecutive interior angles are supplementary.
Converse of Consecutive Interior Angle Theorem
The antipodal of consecutive interior angle theorem states that if a transversal intersects two lines such that a pair of sequent interior angles are supplementary, then the two lines are parallel. The proof of this theorem and its converse is shown beneath.
Referring to the same effigy,
Information technology is given, ∠5 + ∠4 = 180° (Consecutive Interior angles) ---------- (Equation ane)
Since ∠ane and ∠4 form a linear pair of angles,
∠ane + ∠4 = 180° ---------- (Linear pair of angles are supplementary) -----------(Equation 2)
Since the right-hand sides of Equation 1 and Equation ii are equal, we can equate the left-hand side of the equations (1) and (two) and write it as:
∠ane + ∠4 = ∠5 + ∠four
If we solve this, nosotros get ∠1 = ∠5 which forms a corresponding pair in the parallel lines.
Thus, i pair of respective angles is equal in the given figure, which can simply happen if the two lines are parallel. Hence, the converse of consecutive interior angle theorem is proved, which says that if a transversal intersects 2 lines such that a pair of consecutive interior angles are supplementary, then the two lines are parallel.
Consecutive Interior Angles of a Parallelogram
We know that the reverse sides of a parallelogram are parallel, therefore, the consecutive interior angles of a parallelogram are e'er supplementary. Observe the post-obit parallelogram in which ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, and ∠D and ∠A are consecutive interior angles. This tin can be understood as follows:
- If we take AB // CD and BC as the transversal, so ∠B + ∠C = 180°
- If nosotros take AB // CD and AD as the transversal, then ∠A + ∠D = 180°
- If nosotros take AD // BC and CD as the transversal, and then ∠C + ∠D = 180°
- If nosotros take AD // BC and AB as the transversal, then ∠A + ∠B = 180°
Tips on Consecutive Interior Angles
Here are some important points to remember about consecutive interior angles.
- The consecutive interior angles are non-side by side and lie on the same side of the transversal.
- Two lines are parallel if and only if the consecutive interior angles are supplementary.
☛ Related Articles
- Exterior Angle Theorem
- Pairs of Angles
- Supplementary Angles
- Intersecting and Not-Intersecting Lines
Consecutive Interior Angles Examples
-
Instance i: Are the post-obit lines 'fifty' and 'm' parallel?
Solution:
In the given figure, if the angles 125° and 60° are supplementary, so it can exist proved that the lines '50' and '1000' are parallel.
But 125° + threescore° = 185°, which means that 125° and 60° are NOT supplementary.
Thus, as per the Consecutive Interior Angles Theorem, the given lines are Not parallel.
-
Example 2: Use the consecutive interior angles theorem to find the value of angle 'x' if line i and line 2 are parallel.
Solution:
In the effigy, information technology is given that 40° and ∠ten are consecutive interior angles and 'Line i' and 'Line 2' are parallel.
Past the sequent interior angles theorem, ∠x and 40° are supplementary.
∠10 + forty° = 180°
∠x = 180° - 40°Therefore, ∠x = 140°.
go to slidego to slidego to slide
Break down tough concepts through elementary visuals.
Math will no longer be a tough subject, especially when yous understand the concepts through visualizations.
Book a Complimentary Trial Class
Do Questions on Consecutive Interior Angles
go to slidego to slide
FAQs on Sequent Interior Angles
What are Sequent Interior Angles?
Sequent interior angles are formed when a transversal passes through a pair of parallel lines or not-parallel lines. They are formed on the interior sides of the two crossed lines at the signal where the transversal intersects the two lines. If the lines that the transversal crosses are parallel, then, the pair of sequent interior angles are supplementary.
How to Place Consecutive Interior Angles?
Sequent Interior angles can exist identified with the help of the following properties:
- Consecutive interior angles are formed when any ii straight lines are intersected by a transversal.
- They take dissimilar vertices.
- They prevarication between 2 lines.
- They are on the same side of the transversal.
- They share a common side.
- If the consecutive angles are formed betwixt 2 parallel lines cut by a transversal, then they are supplementary.
What is the Consecutive Interior Angles Theorem?
The consecutive interior angles theorem states that if a transversal passes through two parallel lines, information technology makes two pairs of consecutive interior angles that are supplementary. In other words, the consecutive interior angles that are formed by ii parallel lines intersected by a transversal add upwards to 180°.
What is the Converse of Consecutive Interior Angles Theorem?
The converse of the consecutive interior angles theorem states that if a transversal intersects ii lines such that a pair of consecutive interior angles are supplementary, then the 2 lines are parallel.
Are Consecutive Interior Angles Ever Supplementary?
No, sequent interior angles are not e'er supplementary. They are supplementary only when the transversal passes through parallel lines. It is to be noted that consecutive interior angles can also exist formed when a transversal passes through 2 non-parallel lines, although in this instance, they are non supplementary.
Are Sequent Interior Angles Congruent?
Consecutive interior angles are NOT coinciding. They are supplementary if a transversal passes through two parallel lines. It means that they add up to 180°.
What is Another Proper noun for Sequent Interior Angles?
Consecutive interior angles are also known as 'co-interior angles' or 'same-side interior angles'.
What are the Other Angles Formed apart from Sequent Interior Angles when a Transversal Passes Through Two Parallel Lines?
When a transversal passes through two parallel lines, there are other angles formed apart from sequent interior angles, like, corresponding angles, alternating interior angles, alternating exterior angles.
What is the Difference Between Alternate Interior Angles and Consecutive Interior Angles?
The Alternate Interior Angles and Consecutive Interior Angles are different pairs of angles formed when ii parallel lines are cut past a transversal. Alternating interior angles are located betwixt two intersecting lines, but they are on the opposite sides of the transversal. Whereas, Sequent interior angles are located on the within of two lines on the same side of the transversal.
How are Consecutive Interior Angles Related?
If the consecutive interior angles are formed between two parallel lines cutting by a transversal, then they are supplementary. This means they would add together upward to 180°.
How to find Sequent Interior Angles?
We know the rule that if the sequent interior angles are formed betwixt 2 parallel lines cut by a transversal, so they are supplementary. This means if we know one of the consecutive angles in a pair, the other angle can be hands calculated past subtracting information technology from 180°. In other cases, if we have the sequent angles equally (20x + 5)° and (24x - 1)°, we can find the value of x with the following method. Since the consecutive angles are supplementary we can write it every bit (20x + 5)° + (24x - ane)° = 180°. This can be solved equally, 44x + four = 180, and the value of x = four. Now, the value of 10 can be substituted in the given expressions and the consecutive angles will be (20 × 4) + 5 = 85°, and (24 × 4) - 1 = 95°.
Give an example of Consecutive Interior Angles in Real Life.
In existent life, consecutive interior angles can exist seen in various places, for example, in a window grill with vertical and horizontal rods. They are formed when two horizontal rods (two parallel lines) are intersected by a vertical rod (transversal).
What practice Consecutive Interior Angles Look Similar?
Consecutive interior angles grade a figure, somewhat similar the alphabetic character U, where the inner angles are the consecutive interior angles.
How are Consecutive Interior Angles Related to Parallel Lines?
Sequent interior angles are the angles that are formed on the internal side of a transversal when information technology crosses two lines that are parallel. When the transversal passes through two parallel lines, then the consecutive interior angles that are formed are supplementary.
andersonconalothe1964.blogspot.com
Source: https://www.cuemath.com/geometry/consecutive-interior-angles/
0 Response to "From the Following Figure, How Can You Conclude That Lines L and M Are Parallel?"
Post a Comment