In $\Delta ABC $, $ BD \perp AC $ at $ D $ and $\angle DBC = 54^\circ $. $ E $ is a point on $ BC $ such that $\angle CAE = 34^\circ $. What is the measure of $\angle AEB $?

The correct answer is Option B: 70^\circ. In the triangle, $\angle BCD = 90^{\circ} - \angle DBC = 36^{\circ}$. In $\Delta AEC$, using exterior angle or angle sum properties, $\angle AEB = \angle C + \angle CAE = 36^{\circ} + 34^{\circ} = 70^{\circ}$. Step-by-Step Solution & Concepts: Shortcut Trick: In the triangle, $\angle BCD = 90^{\circ} - \angle DBC = 36^{\circ}$. In $\Delta AEC$, using exterior angle or angle sum properties, $\angle AEB = \angle C + \angle CAE = 36^{\circ} + 34^{\circ} = 70^{\circ}$. Formula Used: $$\angle AEB = 180^{\circ} - (\angle EAB + \angle ABE)$$, $$\angle ABE = 90^{\circ} - \angle C$$ Given: In triangle ABC, altitude BD is perpendicular to AC at D. Angle DBC = 54 degrees. E is on BC such that angle CAE = 34 degrees. Steps: Analyze Right Triangle BDC: Since $BD \perp AC$, $\Delta BDC$ is a right-angled triangle at $D$. The sum of angles in $\Delta BDC$ is $180^{\circ}$. Given $\angle DBC = 54^{\circ}$, we find $\angle C = 90^{\circ} - 54^{\circ}$.. Apply Exterior Angle Theorem: Consider $\Delta AEC$. The angle $\angle AEB$ is an exterior angle to the triangle $\Delta AEC$ at vertex $E$. By the Exterior Angle Theorem, an exterior angle is equal to the sum of the two opposite interior angles.. Calculate Final Angle: Substitute the known values $\angle C = 36^{\circ}$ and $\angle CAE = 34^{\circ}$ into the equation. Answer: B is correct because the exterior angle $\angle AEB$ in $\Delta AEC$ is equal to $\angle C + \angle CAE$, which equals $36^{\circ} + 34^{\circ} = 70^{\circ}$. Exam Tip: This problem demonstrates the 'Exterior Angle Theorem', a fundamental property often tested in geometry to relate interior angles of triangles efficiently.

In $\Delta ABC $, $ BD \perp AC $ at $ D $ and $\angle DBC = 5... | Mathematics MCQ | Examoogle

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In $\Delta ABC$ , $BD \perp AC$ at $D$ and $\angle DBC = 54°$ . $E$ is a point on $BC$ such that $\angle CAE = 34°$ . What is the measure of $\angle AEB$ ?

56°

70°

78°

80°

Correct Answer

📐 MA • Math Geometry PropertiesMathematicsPrevious Year Paper 2024

Option B

70°

Quick Summary:

In the triangle, $\angle BCD = 90° - \angle DBC = 36°$ . In $\Delta AEC$ , using exterior angle or angle sum properties, $\angle AEB = \angle C + \angle CAE = 36° + 34° = 70°$ .

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📐MAMath SolutionGeometry Properties

📋 Given

In triangle ABC, altitude BD is perpendicular to AC at D. Angle DBC = 54 degrees. E is on BC such that angle CAE = 34 degrees.

🔢 Formula Used

∠AEB=180°−(∠EAB+∠ABE)\angle AEB = 180° - (\angle EAB + \angle ABE)∠AEB=180°−(∠EAB+∠ABE), ∠ABE=90°−∠C\angle ABE = 90° - \angle C∠ABE=90°−∠C

⚡ Exam Hall Shortcut / Speed Trick

In the triangle, $\angle BCD = 90° - \angle DBC = 36°$ . In $\Delta AEC$ , using exterior angle or angle sum properties, $\angle AEB = \angle C + \angle CAE = 36° + 34° = 70°$ .

⚠️ Common Student Trap / Pitfall

Students often fail to identify the right-angled triangle $\Delta BDC$ to find the value of $\angle C$ first, which is the key to the entire problem.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Analyze Right Triangle BDC

Since $BD \perp AC$ , $\Delta BDC$ is a right-angled triangle at $D$ . The sum of angles in $\Delta BDC$ is $180°$ . Given $\angle DBC = 54°$ , we find $\angle C = 90° - 54°$ .

∠C=36°\angle C = 36°∠C=36°

Apply Exterior Angle Theorem

Consider $\Delta AEC$ . The angle $\angle AEB$ is an exterior angle to the triangle $\Delta AEC$ at vertex $E$ . By the Exterior Angle Theorem, an exterior angle is equal to the sum of the two opposite interior angles.

∠AEB=∠C+∠CAE\angle AEB = \angle C + \angle CAE∠AEB=∠C+∠CAE

Calculate Final Angle

Substitute the known values $\angle C = 36°$ and $\angle CAE = 34°$ into the equation.

∠AEB=36°+34°=70°\angle AEB = 36° + 34° = 70°∠AEB=36°+34°=70°

✅

B is correct because the exterior angle $\angle AEB$ in $\Delta AEC$ is equal to $\angle C + \angle CAE$ , which equals $36° + 34° = 70°$ .

Core Concepts Used

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Exterior Angle Theorem Right-Angled Triangle Properties Sum of Angles in a Triangle

💡 EXAM TIP

This problem demonstrates the 'Exterior Angle Theorem', a fundamental property often tested in geometry to relate interior angles of triangles efficiently.

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