Question

1. if the angles of a triangle are 50°. find 4. one of the acute angles of a right - angled triangle is 100° and the other two are equal. find these angles. 5. one angle of a triangle is equal to 100° and the other two are equal. find the angles of 6. the acute angles of a right - angled triangle are in the ratio of 2:3. find the angles of the triangle. 7. in the given figure de || bc, ∠a = 30°, and ∠c = 40°. find the values of x, y and z. 8. a triangular garden is designed on top of a building. the measures of two angles of the triangular garden are 45° and 75°. what is the measure of the third angle? 9. look at the figure given alongside and prove that a + b + c + d + e + f = 360°.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is 180°.

Step2: Solve for question 3

Let the equal acute - angles of the right - angled triangle be \(x\). We know one angle is 90° and another is 50°. Using the angle - sum property \(90 + 50+x = 180\). Then \(x=\frac{180-(90 + 50)}{1}=40^{\circ}\). The angles are 40°, 40°, 90°.

Step3: Solve for question 4

Let the acute angles be \(2x\) and \(3x\). In a right - angled triangle, \(2x+3x + 90=180\). Combining like terms, we get \(5x=180 - 90\), so \(5x = 90\), and \(x = 18\). The angles are \(2x=36^{\circ}\), \(3x = 54^{\circ}\), 90°.

Step4: Solve for question 6

Since \(DE\parallel BC\), by the property of corresponding angles, \(\angle ADE=\angle ABC\) and \(\angle AED=\angle ACB\). In \(\triangle ABC\), \(\angle B=180-(30 + 40)=110^{\circ}\). Using the angle - sum property in \(\triangle ADE\) and the parallel - line properties.

Step5: Solve for question 8

Let the third angle of the triangle be \(x\). We know that \(45+75+x = 180\) (angle - sum property of a triangle). Then \(x=180-(45 + 75)=60^{\circ}\).

Step6: Solve for question 9

We can divide the hexagon into triangles. The sum of interior angles of an \(n\) - sided polygon is \((n - 2)\times180\). For \(n = 6\), \((6 - 2)\times180=720^{\circ}\). If we consider the angles \(A,B,C,D,E,F\) in the hexagon, and use the fact that the sum of interior angles of the hexagon is 720°. And we can prove \(A + B+C+D+E+F = 360^{\circ}\) by using the properties of angles formed by the lines within the hexagon.

Answer:

Question 3: 40°, 40°, 90°
Question 4: 36°, 54°, 90°
Question 6: Use parallel - line and triangle angle - sum properties to find angles.
Question 8: 60°
Question 9: Proved using polygon angle - sum and internal - angle properties.