Question

1. find the measure of each missing angle.

Explanation:

Step1: Use the property of isosceles right - angled triangles

Since the two small segments on the vertical side and the two small segments on the horizontal side are equal (marked with the same tick marks), the two small right - angled triangles are isosceles. For an isosceles right - angled triangle, the two non - right angles are equal. In the lower isosceles right - angled triangle, \(m\angle1 = 45^{\circ}\) (because in a right - angled isosceles triangle, if the right angle is \(90^{\circ}\), and the sum of angles in a triangle is \(180^{\circ}\), then \(\angle1=\frac{180 - 90}{2}=45^{\circ}\)).

Step2: Use the property of the upper right - angled triangle

In the upper right - angled triangle, one angle is \(35^{\circ}\) and the right angle is \(90^{\circ}\). Using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), let the angle \(\angle2\). Then \(m\angle2=90 - 35=55^{\circ}\) (because in a right - angled triangle, the sum of the two non - right angles is \(90^{\circ}\)).

Step3: Use the property of the large right - angled triangle

In the large right - angled triangle, one non - right angle is \(35^{\circ}+\angle2\). First, \(35^{\circ}+\angle2=35^{\circ}+ 55^{\circ}=90^{\circ}\) (this is wrong, we should use the angle - sum property of the large triangle. The large right - angled triangle has angles \(35^{\circ}+\angle2\), \(\angle3\) and \(90^{\circ}\). Also, from the lower part, we know that \(\angle4=\angle5\) (isosceles triangle property). And \(\angle3+\angle4+\angle5 = 90^{\circ}\). Since \(\angle4=\angle5\), let's start from another way.

In the large right - angled triangle (the whole triangle), one non - right angle is \(35^{\circ}+\angle2\). Wait, better way:
In the large right - angled triangle (right - angled at the bottom left), one of the non - right angles: let's consider the relationship between the angles.
We know that \(\angle3\): In the large right - angled triangle (the whole triangle), if we consider the angle at the top is \(35^{\circ}+\angle2\) (no, wrong approach). Let's use the fact that \(\angle1 = 45^{\circ}\), \(\angle2=55^{\circ}\)
In the large right - angled triangle (the whole triangle), the sum of angles is \(180^{\circ}\). The right angle is \(90^{\circ}\), and one of the non - right angles is \(35^{\circ}+\angle2\) (incorrect). Another approach:
Since \(\angle1 = 45^{\circ}\), in the large triangle (the whole right - angled triangle), the angle adjacent to \(35^{\circ}\) (let's call the vertex angle of the upper small triangle \(A = 35^{\circ}\), the angle \(\angle2\) is adjacent to \(A\) in the vertical side).
The large triangle's non - right angle (at the bottom right) \(\angle3\):
We know that \(\angle3=35^{\circ}\) (because of the property of similar triangles or by observing the angle relationships. The upper small triangle and the large triangle share an angle. The side - angle - side similarity (the two triangles are similar as the vertical and horizontal sides are in proportion (due to the equal tick - marks) and they have a common right - angle - like orientation).

Step4: Use the isosceles triangle property for \(\angle4\) and \(\angle5\)

Since the lower triangle is isosceles (right - angled isosceles, as we found \(\angle1 = 45^{\circ}\)), and \(\angle3 = 35^{\circ}\), and \(\angle3+\angle4+\angle5=90^{\circ}\) (sum of angles in the lower part of the figure where the right angle is divided). Since \(\angle4=\angle5\) (isosceles triangle property, because the two segments on the horizontal side are equal), then \(m\angle4=m\angle5=\frac{90 - 35}{2}=27.5^{\ci…

Answer:

\(m\angle1 = 45^{\circ}\), \(m\angle2 = 55^{\circ}\), \(m\angle3 = 35^{\circ}\), \(m\angle4 = 27.5^{\circ}\), \(m\angle5 = 27.5^{\circ}\)