Question

two similar triangles are shown.
fill in the missing angle measure and missing side length.
(image of two triangles: smaller triangle with side 9 cm, angle 98°, and a missing side; larger triangle with side 18 cm, angle 35°, side 15 cm, and a missing angle)

Explanation:

Step1: Find the missing angle

Similar triangles have corresponding angles equal. First, find the third angle of the larger triangle. The sum of angles in a triangle is \(180^\circ\). So, for the larger triangle, angles are \(35^\circ\), \(98^\circ\) (corresponding to the smaller triangle's \(98^\circ\)), and the missing angle. Wait, no—wait, the smaller triangle has a \(98^\circ\) angle, so the larger triangle must also have a \(98^\circ\) angle? Wait, no, let's check again. Wait, the larger triangle: angles are \(35^\circ\), and the other two? Wait, no, the smaller triangle has a \(98^\circ\) angle, so the larger triangle must have a corresponding \(98^\circ\) angle? Wait, no, maybe I mixed up. Wait, the sum of angles in a triangle is \(180^\circ\). Let's find the missing angle in the larger triangle. Wait, the smaller triangle: one angle is \(98^\circ\), and since they are similar, the larger triangle has a corresponding angle. Wait, the larger triangle has \(35^\circ\), so let's calculate the third angle of the larger triangle. Let the missing angle in the larger triangle be \(x\). Then \(35^\circ + 98^\circ + x = 180^\circ\)? Wait, no, maybe the \(98^\circ\) is in the smaller triangle, so the larger triangle's corresponding angle is \(98^\circ\)? Wait, no, let's re-express. Wait, the two triangles are similar, so corresponding angles are equal. Let's list the angles:

Smaller triangle: angles are \(98^\circ\), and two others. Larger triangle: angles are \(35^\circ\), and two others. Since similar, the angles must match. So, the sum of angles in a triangle is \(180^\circ\). Let's find the missing angle in the larger triangle. Let's assume that the \(98^\circ\) is a corresponding angle. So, in the larger triangle, angles are \(35^\circ\), \(98^\circ\), and the third angle. Wait, no, \(35 + 98 = 133\), so \(180 - 133 = 47\). Wait, but maybe the smaller triangle's angles: one is \(98^\circ\), and the other two correspond to \(35^\circ\) and \(47^\circ\). Wait, maybe I made a mistake. Wait, the problem is to find the missing angle (in the larger triangle, the white box) and the missing side (in the smaller triangle, the white box).

First, angle: sum of angles in a triangle is \(180^\circ\). So, for the larger triangle: angles are \(35^\circ\), and the other two. Wait, the smaller triangle has a \(98^\circ\) angle, so the larger triangle must have a \(98^\circ\) angle (since similar). So, \(35^\circ + 98^\circ + x = 180^\circ\)? No, \(35 + 98 = 133\), so \(x = 180 - 133 = 47^\circ\). Wait, but maybe the smaller triangle's angles: one is \(98^\circ\), another is \(35^\circ\) (corresponding to the larger's \(35^\circ\)), so the third angle is \(180 - 98 - 35 = 47^\circ\). So the missing angle in the larger triangle (the white box) is \(47^\circ\)? Wait, no, maybe the \(98^\circ\) is in the smaller triangle, so the larger triangle's corresponding angle is \(98^\circ\), and the other angle is \(35^\circ\), so the third angle is \(180 - 98 - 35 = 47^\circ\). So the missing angle (in the larger triangle) is \(47^\circ\)? Wait, no, the white box in the larger triangle is the angle, and the white box in the smaller triangle is the side.

Now, side: similar triangles have proportional sides. The larger triangle has sides 18 cm (corresponding to the smaller's 9 cm) and 15 cm (corresponding to the smaller's missing side, let's call it \(y\)). So the ratio of sides is \(18/9 = 2\), so the scale factor is 2. Therefore, the smaller side corresponding to 15 cm is \(15 / 2 = 7.5\) cm? Wait, no, wait: the larger triangle's side is 18 cm, smaller is 9 cm, so scale factor from smaller to larger is 2 (since 9 * 2 = 18). Therefore, the larger's side 15 cm corresponds to the smaller's side \(15 / 2 = 7.5\) cm? Wait, no, wait: if the smaller triangle has a side of 9 cm, and the larger has 18 cm, then the ratio of smaller to larger is \(9/18 = 1/2\). So the smaller triangle's sides are half of the larger's. Therefore, the larger's side is 15 cm, so the smaller's corresponding side is \(15 * (1/2) = 7.5\) cm. Wait, but let's confirm the correspondence. The larger triangle: sides 18 cm (let's say the left side) and 15 cm (the bottom side). The smaller triangle: left side 9 cm, bottom side \(y\) (missing). So the ratio of left sides: 9/18 = 1/2. Therefore, the ratio of bottom sides: \(y/15 = 1/2\), so \(y = 15 * (1/2) = 7.5\) cm.

Wait, but let's check the angles again. The smaller triangle has a \(98^\circ\) angle, so the larger triangle must have a \(98^\circ\) angle (since similar). Then, the larger triangle's angles: 35°, 98°, and \(180 - 35 - 98 = 47°\). So the missing angle (in the larger triangle) is 47°, and the missing side (in the smaller triangle) is 7.5 cm.

Wait, but maybe I mixed up the angle correspondence. Let's re-express:

Similar triangles: corresponding angles are equal. So the smaller triangle has an angle of 98°, so the larger triangle has a corresponding angle of 98°. Then, the larger triangle's other angles: 35° and \(180 - 98 - 35 = 47°\). So the missing angle (in the larger triangle, the white box) is 47°, and the missing side (in the smaller triangle, the white box) is 7.5 cm (or 15/2 = 7.5).

Step2: Verify the side ratio

The ratio of the smaller triangle's side (9 cm) to the larger's (18 cm) is \(9/18 = 1/2\). So the smaller triangle's sides are half the larger's. Therefore, the larger's side of 15 cm corresponds to the smaller's side of \(15 * (1/2) = 7.5\) cm. This checks out.

Step3: Verify the angle sum

For the larger triangle: 35° + 98° + 47° = 180°, which is correct (35 + 98 = 133; 180 - 133 = 47). For the smaller triangle: 98° + 35° + 47° = 180°, which is also correct. So the angles are consistent.

Answer:

Missing angle (larger triangle): \(47^\circ\)
Missing side (smaller triangle): \(7.5\) cm