Calculate the size of each lettered angle in the following figures : 32
The Correct Answer and Explanation is:
Here is the step-by-step calculation for each lettered angle in the figures provided.
Figure (i)
Analysis of the Diagram
Let’s label the vertices of the large triangle as A (top left), B (top right), and C (bottom). Let the internal point be D.
From the diagram, we are given:
- The angle at vertex B, ∠ABC = 32°.
- The markings indicate two pairs of equal sides:
- Side AB = Side AD (indicated by single tick marks).
- Side CB = Side CD (indicated by double tick marks).
- The lettered angles are x = ∠BAC, y = ∠BCA, and z = ∠ADC.
The quadrilateral ABDC has two pairs of equal-length sides that are adjacent to each other (AB=AD and CB=CD). This is the definition of a kite.
Step-by-Step Calculation:
- Find the angle z:
A key property of a kite is that the angles between the pairs of unequal sides are equal. In the kite ABDC, the pairs of unequal sides are (AB, BC) and (AD, DC). The angles between them are ∠ABC and ∠ADC.
Therefore, ∠ADC = ∠ABC.
We are given ∠ABC = 32°.
So, z = ∠ADC = 32°. - Find the relationship between x and y:
Now consider the large triangle ΔABC. The sum of the angles in a triangle is 180°.
∠BAC + ∠ABC + ∠BCA = 180°
Substituting the known values and variables:
x + 32° + y = 180°
x + y = 180° – 32°
x + y = 148°
Conclusion for Figure (i):
Based on the information provided in the diagram, we can determine that z = 32°. However, we can only find the sum of angles x and y (x + y = 148°), not their individual values. There might be missing information in the problem statement to solve for x and y uniquely. For example, if it were also given that the triangle ABC is isosceles, we could solve it completely.
Figure (ii)
Analysis of the Diagram
Let’s label the vertices of the large triangle as A (top), B (bottom left), and C (bottom right). A point E is on side AB, and a point D is on side AC. The line segments BD and CE intersect at point F.
From the diagram, we are given:
- The angle at vertex A, ∠BAC = 35°.
- The markings indicate several equal side lengths:
- Side AB = Side AC (indicated by double tick marks).
- Side AE = Side AD (indicated by single tick marks).
- The lettered angles are x = exterior angle at C, y = ∠FBC (or ∠DBC), and z = ∠BFC.
Step-by-Step Calculation:
- Analyze the large triangle ΔABC:
Since AB = AC, ΔABC is an isosceles triangle. The base angles opposite to the equal sides must be equal.
∠ABC = ∠ACB.
The sum of angles in ΔABC is 180°.
∠BAC + ∠ABC + ∠ACB = 180°
35° + 2 * ∠ACB = 180°
2 * ∠ACB = 180° – 35° = 145°
∠ACB = 145° / 2 = 72.5°.
So, ∠ABC = ∠ACB = 72.5°. - Find the angle x:
Angle x is on a straight line with ∠ACB.
x + ∠ACB = 180°
x + 72.5° = 180°
x = 180° – 72.5° = 107.5°. - Prove Congruency and Parallel Lines:
Consider triangles ΔABD and ΔACE.- AB = AC (Given)
- ∠BAD = ∠CAE (Common angle ∠A = 35°)
- AD = AE (Given)
By the Side-Angle-Side (SAS) congruence rule, ΔABD ≅ ΔACE.
A consequence of this is that corresponding angles are equal: ∠ABD = ∠ACE.
- Analyze the internal triangles:
From ∠ABC = ∠ACB and ∠ABD = ∠ACE, we can deduce:
∠ABC – ∠ABD = ∠ACB – ∠ACE
∠DBC = ∠ECB
Since y = ∠DBC = ∠FBC and ∠ECB = ∠FCB, we have ∠FBC = ∠FCB = y.
This means that ΔFBC is an isosceles triangle with sides FB = FC.
The sum of angles in ΔFBC is 180°.
∠FBC + ∠FCB + ∠BFC = 180°
y + y + z = 180°
2y + z = 180° - Use a property of similar triangles:
Consider ΔADE. Since AE = AD, it is an isosceles triangle.
∠ADE = ∠AED = (180° – 35°) / 2 = 145° / 2 = 72.5°.
Notice that ∠ADE = ∠ABC = 72.5°. These are corresponding angles with respect to the transversal line AB. Therefore, the line segment DE is parallel to BC. - Find the values of y and z:
Since DE || BC, we can use the property of alternate interior angles with transversal BD.
∠BDE = ∠DBC = y.
Now consider the sum of angles in ΔABD, which is 180°.
∠BAD + ∠ABD + ∠ADB = 180°.
We know ∠ADB = ∠ADE + ∠EDB = 72.5° + y.
Also, ∠ABD = ∠ABC – ∠DBC = 72.5° – y.
Substituting these into the sum of angles for ΔABD:
35° + (72.5° – y) + (72.5° + y) = 180°
35° + 145° = 180°
180° = 180°
This confirms our geometric relationships are correct, but it does not provide a unique solution for y and z. Similar to the first problem, it seems there is not enough information to determine y and z individually.
Conclusion for Figure (ii):
We can determine that x = 107.5°. We can also establish a firm relationship between y and z: z = 180° – 2y. However, the problem as stated does not provide sufficient constraints to calculate unique values for y and z.thumb_upthumb_down
