Geometry Problem: Angle Sum in Triangle and Quadrilateral
Unfortunately, the image is partially cropped, so the full text of the problem isn't visible. However, I can see the first two parts of the question which I can help you with:
(i) For the triangle, show that \( 2a + 5b = 171 \)
(ii) For the quadrilateral, show that \( 7a + 8b = 342 \)
Let's solve each part step by step:
**Part (i): For the triangle**
In any triangle, the sum of the interior angles is always 180 degrees. The triangle shown in the image has angles \( 5b, a + b \), and \( a - 29 \).
To show that \( 2a + 5b = 171 \), we'll add up the angles and set them equal to 180 degrees, which is the sum of all interior angles of a triangle.
\[ 5b + (a + b) + (a - 29) = 180 \]
\[ 5b + a + b + a - 29 = 180 \]
\[ 2a + 6b - 29 = 180 \]
We add \( 2a + 6b \) and subtract 29 to isolate \( 2a + 5b \) on one side:
\[ (2a + 6b) - 29 = 180 \]
\[ 2a + 6b = 180 + 29 \]
\[ 2a + 6b = 209 \]
However, to get \( 2a + 5b \), we need to subtract \( b \) from both sides:
\[ (2a + 6b) - b = 209 - b \]
\[ 2a + 5b = 209 - b \]
Since our goal is to show \( 2a + 5b = 171 \), this means b must be \( 209 - 171 = 38 \). However, without additional information or an equation for b, we cannot verify this from the cropped image. There might be more information on the paper that provides a way to determine the value of b.
**Part (ii): For the quadrilateral**
The sum of the interior angles of a quadrilateral is \( 360 \) degrees. The quadrilateral shown in the image has angles \( 3a + b, a, 3a + 2b \), and \( 3b \).
We'll add all angles and set them equal to 360 degrees:
\[ (3a + b) + a + (3a + 2b) + 3b = 360 \]
\[ 3a + b + a + 3a + 2b + 3b = 360 \]
\[ 7a + 6b = 360 \]
This is slightly different from the expression you were asked to show, \( 7a + 8b = 342 \). It seems there might be an error in the transcription of the problem or more information is needed that isn't visible due to the partial cropping of the image.
In any case, if you have more details or can provide additional information or the rest of the image, I can assist you further.
