CBSE Class 7 Mathematics Lines and Angles Assignment Set 02

Question 1. In the given figure, \(EF \parallel GH\), \(\angle EAB = 65^\circ\) and \(\angle ACH = 100^\circ\). Find the value of \(\angle ABC\), \(\angle ACB\) and \(\angle CAF\).
Answer:
Given lines \(EF \parallel GH\), transversal \(AC\) intersects them at points \(A\) and \(C\).
1. Finding \(\angle CAF\):
\(\angle EAB\) and \(\angle CAF\) are vertically opposite angles.
Therefore, \(\angle CAF = \angle EAB = 65^\circ\).
2. Finding \(\angle ACB\):
\(\angle EAB\) and \(\angle ACH\) are alternate interior angles because \(EF \parallel GH\) and \(AC\) is the transversal.
So, \(\angle ACH = \angle EAB + \angle CAB\) (looking at the angle addition around point \(A\), or directly using parallel line properties).
Alternatively, since \(EF \parallel GH\), alternate interior angle to \(\angle EAB\) is \(\angle GAC\) or similar, let us use straight line properties:
\(\angle EAB + \angle BAC + \angle CAF = 180^\circ\) (angles on a straight line \(EF\)).
\(65^\circ + \angle BAC + 65^\circ = 180^\circ\)
\(\angle BAC = 180^\circ - 130^\circ = 50^\circ\).
Now, using alternate interior angles for parallel lines \(EF \parallel GH\) with transversal \(AC\):
\(\angle FAC\) and \(\angle ACG\) are alternate interior angles. Thus, \(\angle ACG = \angle FAC = 65^\circ\).
Since \(\angle ACH = 100^\circ\) and \(\angle ACH = \angle ACG + \angle GCB\)... wait, looking at the figure, line \(GH\) is a straight line containing points \(G\), \(B\), \(C\), \(H\).
Given \(\angle ACH = 100^\circ\). Since \(GH\) is a straight line, \(\angle ACB + \angle ACH = 180^\circ\) (linear pair).
\(\angle ACB = 180^\circ - 100^\circ = 80^\circ\).
3. Finding \(\angle ABC\):
In \(\triangle ABC\), the sum of angles is \(180^\circ\).
\(\angle BAC + \angle ACB + \angle ABC = 180^\circ\)
\(50^\circ + 80^\circ + \angle ABC = 180^\circ\)
\(130^\circ + \angle ABC = 180^\circ\)
\(\angle ABC = 180^\circ - 130^\circ = 50^\circ\).
Thus, \(\angle ABC = 50^\circ\), \(\angle ACB = 80^\circ\), and \(\angle CAF = 65^\circ\).
In simple words: We used linear pairs, vertically opposite angles, and the angle sum property of a triangle to find the unknown angles.

Exam Tip: Always identify parallel lines and transversals first to easily spot alternate interior and corresponding angles.

Question 2. In the given figure, \(l \parallel m\) and \(t\) is a transversal. If \(\angle 5 = 70^\circ\), find the measure of other angles.
Answer:
Given \(l \parallel m\) with transversal \(t\) and \(\angle 5 = 70^\circ\).
1. \(\angle 4\) and \(\angle 5\) form a linear pair (assuming standard labeling in such parallel line diagrams):
Looking at the standard configuration where \(\angle 5\) is an exterior/interior angle on transversal \(t\):
By vertically opposite angles: \(\angle 8 = \angle 5 = 70^\circ\) (if \(\angle 5\) and \(\angle 8\) are vertically opposite) or using corresponding/alternate angles:
Let us determine each angle systematically based on parallel lines properties:
- \(\angle 5 = 70^\circ$
- \(\angle 4\) is vertically opposite to \(\angle 5\) (or adjacent linear pair depending on exact diagram, but typically in these figures, numbered 1 to 8):
Let us assume standard interior/exterior alternate positions:
\(\angle 2 = \angle 5 = 70^\circ\) (alternate interior angles, if \(\angle 2\) and \(\angle 5\) are alternate)
Let us calculate all angles directly using standard rules for \(l \parallel m\):
- If \(\angle 5 = 70^\circ\), then its linear pair is \(180^\circ - 70^\circ = 110^\circ\).
- Corresponding angles are equal: \(\angle 1 = \angle 5 = 70^\circ\) (or corresponding pair).
- Alternate interior angles are equal.
- Vertically opposite angles are equal: \(\angle 3 = 70^\circ\), \(\angle 6 = 110^\circ\), \(\angle 7 = 110^\circ\), \(\angle 8 = 70^\circ\) (matching standard angle positions around the two intersections).
In simple words: Once you know one angle between parallel lines and a transversal, all other seven angles can be found using linear pairs (\(180^\circ\) minus the angle) or equality (alternate/corresponding/vertically opposite angles).

Exam Tip: Remember the rule: Out of the 8 angles formed, 4 are acute and equal to each other, and 4 are obtuse and equal to each other.

Question 3. From the given figure, find the measure of \(\angle ECD\) where \(AO \parallel CD\), \(OB \parallel CE\).
Answer:
Given:\ - \(AO \parallel CD\)
- \(OB \parallel CE\)
- \(\angle AOB = 50^\circ\)
Since \(AO \parallel CD\) and \(OB\) acts as a transversal or line segment connecting the angles:
By corresponding angles or parallel line properties between the arms of the angles:
The angle between two lines is equal to the angle between their respective parallel lines (\(\angle ECD = \angle AOB\)).
Therefore, \(\angle ECD = 50^\circ\).
In simple words: If two angles have their arms parallel to each other, the angles are either equal or supplementary. Here, \(\angle ECD = \angle AOB = 50^\circ\).

Exam Tip: If the arms of one angle are parallel to the arms of another angle respectively, the two angles are equal.

Question 4. Find the values of \(x\) and \(y\), where \(l \parallel m\).
Answer:
1. Finding \(x\):
The angle \(50^\circ\) and the angle \(x^\circ\) are interior angles on the same side of the transversal (consecutive interior angles) between parallel lines \(l \parallel m\).
Consecutive interior angles are supplementary (sum to \(180^\circ\)).
\(x^\circ + 50^\circ = 180^\circ\)
\(x = 180 - 50 = 130\).
2. Finding \(y\):
Similarly, the angle \(65^\circ\) and the angle \(y^\circ\) are consecutive interior angles between the two parallel lines.
\(y^\circ + 65^\circ = 180^\circ\)
\(y = 180 - 65 = 115\).
Thus, \(x = 130\) and \(y = 115\).
In simple words: Co-interior angles between parallel lines add up to \(180^\circ\). Subtract the given angle from \(180^\circ\) to find the unknown variable.

Exam Tip: Co-interior angles (C-angles) always add up to \(180^\circ\) when lines are parallel.