(i) The measures of the angles are m ∠ TCM = m ∠ TCQ = m ∠ CMQ = 90°.
(ii) The lengths of sides CM and CQ are 2 cm and 2√2 cm, respectively.
(iii) The angle between the <em>side</em> face and the base is equal to θ = cos⁻¹ (2 / MT) and the angle between the <em>slant</em> edge and the base is θ = cos⁻¹ [2√2 / √(4 + MT²)].
<h3>How to analyze a right pyramid</h3>
In this problem we have a <em>right</em> pyramid as line segment TC is perpendicular to the base PQRS, which means that any line coplanar with PQRS is perpendicular to line segment TC. i) the measures of the angles are m ∠ TCM = m ∠ TCQ = 90°.
Besides, the base PQRS is a square, which is equivalent to four right triangles with angles 45° - 45° - 90° and each triangle can be divided into two <em>right</em> triangles. Hence, the measure of the angle CMQ is 90°.
ii) The length of the sides can be found by the 45-45-90 theorem, which states that the length of the hypotenuse is √2 times the length of any of the legs:
CQ = (√2 / 2) · PQ
CQ = (√2 / 2) · (4 cm)
CQ = 2√2 cm
CM = (√2 / 2) · (2 √2 cm)
CM = 2 cm
iii) The angle between the <em>side</em> face, one of them contains the <em>line</em> segment MT and the base can be found by the following <em>inverse trigonometric</em> relation: (value of the pyramid height is missing)
θ = cos⁻¹ (CM / MT)
θ = cos⁻¹ (2 / MT) (1)
Similarly, the angle between the <em>slant</em> edge and the base is found by Pythagorean theorem and <em>inverse trigonometric</em> relation: (value of the pyramid height is missing)
θ = cos⁻¹ (CQ / TQ)
θ = cos⁻¹ [2√2 / √(QM² + MT²)]
θ = cos⁻¹ [2√2 / √(4 + MT²)] (2)
To learn more on pyramids: brainly.com/question/17615619
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