If two lines AB and CD intersect at O such that ∠AOC = 5 ∠AOD, then the four angles at O are
30°, 30°, 150°, 150°
Understanding Angles Formed by Intersecting Lines
When two lines intersect at a point, they form four angles around that point. These angles have specific relationships, such as forming linear pairs and being vertically opposite to each other. This problem involves finding the measures of these four angles given a relationship between two adjacent angles.
Key Angle Concepts
To solve this problem, we need to recall two important geometric concepts:
- Linear Pair: Two adjacent angles that form a straight line. The sum of angles in a linear pair is always \(180^\circ\).
- Vertically Opposite Angles: When two lines intersect, the angles opposite each other at the intersection point are called vertically opposite angles. Vertically opposite angles are always equal in measure.
Solving the Problem: Finding the Angles
We are given two lines AB and CD that intersect at point O. The angles formed at O are \(\angle AOC\), \(\angle AOD\), \(\angle BOD\), and \(\angle BOC\).
We are given the relationship: \(\angle AOC = 5 \angle AOD\).
Let's denote \(\angle AOD\) as \(x\). According to the given information, \(\angle AOC = 5x\).
Using the Linear Pair Property
Angles \(\angle AOC\) and \(\angle AOD\) form a linear pair on the line CD (or AB). Therefore, their sum is \(180^\circ\).
\(\angle AOC + \angle AOD = 180^\circ\)
Substitute the expressions in terms of \(x\):
\(5x + x = 180^\circ\)
Combine the terms:
\(6x = 180^\circ\)
Solve for \(x\):
\(x = \frac{180^\circ}{6}\)
\(x = 30^\circ\)
Calculating the Measures of Angles
Now that we have the value of \(x\), we can find the measures of \(\angle AOD\) and \(\angle AOC\):
- \(\angle AOD = x = 30^\circ\)
- \(\angle AOC = 5x = 5 \times 30^\circ = 150^\circ\)
Using the Vertically Opposite Angles Property
Now we can find the measures of the other two angles using the property of vertically opposite angles:
- \(\angle BOD\) is vertically opposite to \(\angle AOC\). Therefore, \(\angle BOD = \angle AOC = 150^\circ\).
- \(\angle BOC\) is vertically opposite to \(\angle AOD\). Therefore, \(\angle BOC = \angle AOD = 30^\circ\).
Summary of the Four Angles
The four angles formed at the intersection point O are:
- \(\angle AOC = 150^\circ\)
- \(\angle AOD = 30^\circ\)
- \(\angle BOD = 150^\circ\)
- \(\angle BOC = 30^\circ\)
Listing the four angles in numerical order gives: \(30^\circ, 30^\circ, 150^\circ, 150^\circ\).
| Angle | Relation | Measure |
|---|---|---|
| \(\angle AOD\) | Let it be \(x\) | \(30^\circ\) |
| \(\angle AOC\) | \(5 \angle AOD\) | \(150^\circ\) |
| \(\angle BOC\) | Vertically opposite to \(\angle AOD\) | \(30^\circ\) |
| \(\angle BOD\) | Vertically opposite to \(\angle AOC\) | \(150^\circ\) |
These angles \(30^\circ, 30^\circ, 150^\circ, 150^\circ\) correspond to one of the given options.
Revision Table: Key Geometric Properties
| Property | Description | Relationship |
|---|---|---|
| Linear Pair | Two adjacent angles on a straight line | Sum is \(180^\circ\) |
| Vertically Opposite Angles | Angles opposite at the intersection of two lines | Angles are equal |
| Angles Around a Point | Sum of all angles around a single point | Sum is \(360^\circ\) |
Additional Information on Intersecting Lines and Angles
When lines intersect, they create pairs of angles with specific properties. Understanding these properties is crucial for solving geometry problems involving intersecting lines. Apart from linear pairs and vertically opposite angles, other angle relationships exist when parallel lines are involved, such as corresponding angles, alternate interior angles, and consecutive interior angles. However, for simple intersecting lines without parallelism, linear pairs and vertically opposite angles are the primary concepts.
Remember that the sum of all four angles around the intersection point O must be \(360^\circ\). Let's check our result: \(30^\circ + 150^\circ + 30^\circ + 150^\circ = 360^\circ\). This confirms our calculations are correct.
Similar Questions
In the given figure AB is parallel to CD and AC is parallel to BD. If ∠EAC = 40°, ∠FDG = 55°, ∠HAB = x°, then what is the value of x?
In a pi-diagram there are three sectors. If the ratio of the angles of the sectors is 1 : 2 : 3, then what is the angle of the largest sector?
In the given figure, if \(\frac{y}{x} = 6\) and \(\frac{z}{x} = 5\) , then what is the value of x?
Angles are shown in the given figure. What is value of ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8?
In the given figure PQ is parallel to RS, ∠AEF = 95°, ∠BHS = 110°, and ∠ABC = x°. Then what is the value of x?
Three parallel lines x, y and z are cut by two transversals m and n. Transversal m cuts the lines, x, y, z at P, Q, R respectively, and transversal n cuts the lines x, y, z at L, M, N respectively. If PQ = 3 cm, QR = 9 cm and MN = 10.5cm, then what is the length of LM?
The locus of a point equidistant from two intersecting lines is
In the figure given below, p, q, r are parallel lines; l and m are two transversals.
Consider the following:
1. AB : AC = DE : DF
2. AB × EF = BC × DE
Which of the above is/are correct?
The length of a line segment AB is 2 cm. It is divided into two parts at a point C such that AC 2= AB × CB. What is the length of CB?
Consider the following statements in respect of three straight lines A, B and C on a plane:
1. If A and C are parallel and B and C are parallel, then A and B are parallel
2. If A is perpendicular to C and B is perpendicular to C, then A and B are parallel
3. If the acute angle between A and C is equal to the acute angle between B and C; then A and B are parallel
Which of the above statements are correct?
Important Questions from Lines and Angles
Two parallel lines are intersected by a transversal, the corresponding angles are:
If l, m, n are lines such that, l is parallel to n and m is parallel to n, then, ______
There are three points P, Q and R on a straight line such that PQ : QR = 3 : 5. If n is the number of possible values of PQ : PR, then what is n equal to ?
Two cars start from a point at the same time and travel on two different roads at right angles to each other. Their speed is 18 km/h and 72 km/h respectively. What will be the distance between them after 6 seconds?
If angles of a triangle are in the ration of 2 : 3 : 4, then the measure of the smallest angle is:
