Given the image, calculate the missing angle in triangle ABC, where angle C is 90 degrees, angle B is unknown, angle BAC is 70 degrees, and the side AC has length 18. Additionally, consider a second triangle where angle C is 90 degrees, angle B is unknown, and an exterior angle at vertex B is 124 degrees. Calculate angle BAC for this second triangle.

Let's break down these geometry problems step by step!

Problem 1: Triangle ABC

We have a right-angled triangle ABC, where:

• Angle C = 90 degrees
• Angle BAC = 70 degrees
• Angle ABC is unknown

The sum of angles in any triangle is always 180 degrees.

So, Angle BAC + Angle ABC + Angle C = 180 degrees

\[ 70^\circ + \text{Angle ABC} + 90^\circ = 180^\circ \]

\[ \text{Angle ABC} = 180^\circ - 90^\circ - 70^\circ \]

\[ \text{Angle ABC} = 20^\circ \]

Problem 2: Second Triangle

In the second triangle, we have:

• Angle C = 90 degrees
• An exterior angle at vertex B = 124 degrees
• Angle BAC is unknown

The exterior angle and the interior angle at a vertex are supplementary, meaning they add up to 180 degrees.

So, Interior Angle ABC + Exterior Angle at B = 180 degrees

\[ \text{Interior Angle ABC} + 124^\circ = 180^\circ \]

\[ \text{Interior Angle ABC} = 180^\circ - 124^\circ \]

\[ \text{Interior Angle ABC} = 56^\circ \]

Now, we can find Angle BAC using the sum of angles in a triangle:

Angle BAC + Angle ABC + Angle C = 180 degrees

\[ \text{Angle BAC} + 56^\circ + 90^\circ = 180^\circ \]

\[ \text{Angle BAC} = 180^\circ - 90^\circ - 56^\circ \]

\[ \text{Angle BAC} = 34^\circ \]

Ответ: В первом треугольнике угол B равен 20 градусам. Во втором треугольнике угол A равен 34 градусам.