Step 1: Analyze Triangle BDC
Identify known angles in the right-angled triangle $\triangle BDC$.
- Given $BD \perp AC$, meaning $\angle BDC = 90^\circ$.
- Given $\angle DBC = 40^\circ$.
- Calculate $\angle BCD$ using the angle sum property ($180^\circ$) for $\triangle BDC$: $\angle BCD = 180^\circ - 90^\circ - 40^\circ = 50^\circ$.
- Therefore, $\angle BCA = 50^\circ$.
Step 2: Analyze Triangle AEC
Use the known angle $\angle BCA$ and the given angle $\angle CAE$ in $\triangle AEC$.
- The angle $\angle ACE$ is the same as $\angle BCA$, so $\angle ACE = 50^\circ$.
- Given $\angle CAE = 37^\circ$.
- Calculate $\angle AEC$ using the angle sum property ($180^\circ$) for $\triangle AEC$: $\angle AEC = 180^\circ - (\angle CAE + \angle ACE) = 180^\circ - (37^\circ + 50^\circ) = 180^\circ - 87^\circ = 93^\circ$.
Step 3: Calculate Angle AEB
Determine $\angle AEB$ using the relationship between $\angle AEB$ and $\angle AEC$.
- The point E lies on the line segment BC.
- Angles $\angle AEB$ and $\angle AEC$ form a linear pair along the line BC at point E. They are supplementary angles.
- Thus, $\angle AEB + \angle AEC = 180^\circ$.
- Calculate $\angle AEB$: $\angle AEB = 180^\circ - \angle AEC = 180^\circ - 93^\circ = 87^\circ$.
Final Answer
The measure of angle $\angle AEB$ is $87^\circ$.
Similar Questions
- In $\Delta ABC$, $BD \perp AC$ at D and $\angle DBC = 71^\circ$. E is a point on BC such that $\angle CAE = 17^\circ$. What is the measure of $\angle AEB$?
- In triangle ABC, bisector of $\angle\text{ABC}$ and $\angle\text{ACB}$ meet at O. If $\angle\text{BAC} = 60^\circ$, then find the measure of $\angle\text{BOC}$.
- In $\Delta\text{XYZ}$, $\text{XY} = 12\text{ cm}$ and $\text{YZ} = 18\text{ cm}$. $\text{XW}$, the angle bisector of $\text{YXZ}$, meets $\text{YZ}$ at $\text{W}$, such that $\text{YW} : \text{WZ}$ is $4 : 5$. Find the length of the third side of the triangle.
- In $\Delta\text{PQR}$, $\text{QR}$ is extended up to $\text{S}$ so that $\text{RS} = \text{RP}$. If $\angle\text{PRQ} = 70^\circ$ and $\angle\text{QPS} = 110^\circ$ then find the measure of $\angle\text{PQS}$.
- The areas of two similar triangles are respectively $16\text{ m}^{2}$ and $36\text{ m}^{2}$. Find the ratio of their corresponding sides.
- In $\Delta PQR$, QR is extended up to S, so that RS = RP. If $\angle RPQ = 55^\circ$ and $\angle PRS = 110^\circ$, then find the measure of $\angle PQS$.
- The sides of a triangle are 44 cm, 33 cm, and 55 cm. What is its area? (in $cm^2$)
- The perimeter of an equilateral triangle ABC is 22.2 cm. What is the area of triangle (in $cm^2$)?
- In $\triangle\text{ABC}$, $\text{BD} \perp \text{AC}$ at $\text{D}$ and $\angle\text{DBC} = 60^{\circ}$. $\text{E}$ is a point on $\text{BC}$ such that $\angle\text{CAE} = 20^{\circ}$. What is the measure of $\angle\text{AEB}$?
- In a triangle, if angle $A = 30^\circ$ and angle $B = 45^\circ$, then what is the angle of C?
Important Questions from Triangles
What is the circumcenter of the triangle ABC?
What is the centroid of the triangle ABC?
What is the foot of the altitude from the vertex A of the triangle ABC?
In ΔABC, D is a point on BC such that ∠ADB = 2∠DAC, ∠BAC = 70° and ∠B = 56°. What is the measure of ∠ADC?
In ΔABC, ∠A = 66° and ∠B = 50 °. If the bisectors of ∠B and ∠C meet at P, then ∠BPC – ∠PCA = ?
