Question

In the given figure, \(\Delta ODC \sim \Delta OBA\). If \(\angle BOC = 110^\circ\), \(\angle ODC = 45^\circ\) and \(AB = 2 CD\), then find (i) \(m\angle OAB\) (ii) \(OB : OD\).

Hint:

Always write out the similarity statement carefully.
The order of letters in \(\Delta ODC \sim \Delta OBA\) tells you exactly which vertices correspond to each other:
\(O \leftrightarrow O\), \(D \leftrightarrow B\), and \(C \leftrightarrow A\).
This makes finding corresponding angles and side ratios highly systematic.

Answer 1

Correct Answer: 1

Step 1: Understanding the Question:
We are given two similar triangles: \(\Delta ODC \sim \Delta OBA\).
We are also given:
- \(\angle BOC = 110^\circ\)
- \(\angle ODC = 45^\circ\)
- \(AB = 2 CD\)
We need to determine (i) the measure of angle \(\angle OAB\) and (ii) the ratio of lengths \(OB : OD\).

Step 2: Key Formula or Approach:
1. Since the line segment \(BD\) is a straight line, the angles \(\angle DOC\) and \(\angle BOC\) form a linear pair:
\[ \angle DOC + \angle BOC = 180^\circ \]
2. In any triangle, the sum of all interior angles is \(180^\circ\). We will apply this to \(\Delta ODC\).
3. When two triangles are similar, their corresponding angles are equal, and their corresponding sides are in the same proportion.
\[ \Delta ODC \sim \Delta OBA \implies \angle OAB = \angle OCD \]
And for the sides:
\[ \frac{OB}{OD} = \frac{AB}{CD} \]

Step 3: Detailed Explanation:

• Let us first find the angle \(\angle DOC\). Since \(BD\) is a straight line passing through \(O\):
\[ \angle DOC + \angle BOC = 180^\circ \text{ (Linear pair)} \]
\[ \angle DOC + 110^\circ = 180^\circ \]
\[ \angle DOC = 180^\circ - 110^\circ = 70^\circ \]

• Now, look at \(\Delta ODC\). The sum of angles in this triangle must be \(180^\circ\):
\[ \angle ODC + \angle DOC + \angle OCD = 180^\circ \]
\[ 45^\circ + 70^\circ + \angle OCD = 180^\circ \]
\[ 115^\circ + \angle OCD = 180^\circ \]
\[ \angle OCD = 180^\circ - 115^\circ = 65^\circ \]

• Since \(\Delta ODC \sim \Delta OBA\), the corresponding angles of similar triangles are equal:
- \(\angle OAB\) corresponds to \(\angle OCD\).
- Therefore:
\[ m\angle OAB = \angle OCD = 65^\circ \]
This answers part (i).

• For part (ii), use the ratio of corresponding sides from the similarity \(\Delta ODC \sim \Delta OBA\):
\[ \frac{OB}{OD} = \frac{AB}{CD} \]

• We are given in the problem statement that \(AB = 2CD\). Substitute this into the ratio:
\[ \frac{OB}{OD} = \frac{2CD}{CD} = 2 \]
- Therefore, the ratio \(OB : OD\) is \(2 : 1\).


Step 4: Final Answer:
(i) \(m\angle OAB = 65^\circ\)
(ii) \(OB : OD = 2 : 1\)